Integrate a rational function of x/sqrt(P) for an even quartic P, high powers of two linears, and nine more algebraic shapes (#2023)

Eleven shapes of Rubi's family 1, each answered where the rules declined or
ran out of time:
- a rational function beside a half-odd power of P = a + b x^2 + c x^4,
  where it is one of t = x/sqrt(P), since dt/dx = (a - c x^4)/P^(3/2);
- (c x)^t, t not whole, as x^t beside (c x)^t x^(-t), constant wherever
  defined;
- two roots whose radicands' quotient is a constant times an even power of
  x, as one root up to a factor whose square is constant;
- a polynomial over high whole powers of two linears, under t = L1/L2;
- an algebraic function of 1/x alone, under x = 1/u;
- x^m (a + b x^n)^p, m and n whole and p <= -2, by the reduction to p = -1;
- a power of a sum of powers of x beside a symbolic power of a binomial,
  multiplied out into binomial differentials;
- a difference of radicands that is one power of x, written as one;
- a rational function beside an odd root of a linear past the square,
  split first where a linear below meets the root at an irrational value;
- an odd integrand beside a high power of a quadratic left to u = x^2, and
  a quartic in one shifted linear to u = d + k x.

Chain against master f7f2a7a: pocket of 1,079 rows of family 1,
1,013 -> 1,038 solved, 0 wrong; samples unchanged but family 1's 40-a-file,
1,375 -> 1,379; the 25 rows that moved, alone, 0 -> 25 solved, each in
18-772 ms; no row's allocation grew past 1.3x.
x^2/((1 - a x)^16 (1 + a x)^11), which master answers in 20 s and 8.5 GB as
16,292 characters (exactly right; the probe's double evaluation calls it
wrong), is 1,380 in 0.12 s.

Part of #233.


Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

Co-authored-by: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

Author: Rafael-SOWNet

BREAKING-CHANGES.md

@@ -3028,6 +3028,46 @@ without a common factor
 | `"(c + d*tan(x))^(3/2)/(a + i*a*tan(x))^2".ToEntity().Integrate("x")` | `integral(...)`; 470,119 characters on the unreleased master | 77,377 characters |
 | `"(c + d*tan(x))^(1/2)/(a + i*a*tan(x))^2".ToEntity().Integrate("x")` | `integral(...)`; 258,943 characters on the unreleased master | 71,266 characters |
 
+### A rational function of `x/sqrt(P)` for an even quartic `P`, high powers of two linears, and nine more algebraic shapes are integrated
+
+**Answers where there were none, and shorter ones.** Rubi's algebraic integrands of these shapes were declined or ran
+out of time ([#233](https://github.com/asc-community/AngouriMath/issues/233)):
+- A rational function beside a half-odd power of `P = a + b x^2 + c x^4`, where it is a rational function of
+  `t = x/sqrt(P)`: `dt/dx = (a - c x^4)/P^(3/2)`, so `(a - c x^4)/((a d + a e x^2 + c d x^4) sqrt(P))` is
+  `dt/(d + (a e - b d) t^2)`. Which integrands are is read in `y = x^2` modulo `y^2 - w y + a/c`, `w = y + a/(c y)`;
+  an elliptic integral does not pass.
+- `(c x)^t`, `t` not whole, written `x^t` beside `(c x)^t x^(-t)`, whose derivative is zero wherever it is defined:
+  `(c x)^(-1 - 2 n - n p) (a + b x^n)^p`.
+- Two square roots whose radicands have a constant times an even power of x for their quotient are one root, up to a
+  factor whose square is constant: `(g + h x) sqrt(a + b x + c x^2)/(a d + b d x + c d x^2)^(3/2)`.
+- A polynomial over high whole powers of two linears, under `t = L1/L2`: `x^2/((1 - a x)^16 (1 + a x)^11)`.
+- An algebraic function of `1/x` alone, under `x = 1/u`: `1/((a + b/x)^(5/2) (c + d/x)^2)`.
+- `x^m (a + b x^n)^p` for whole `m` and `n` and `p ≤ -2`, by the reduction that raises `p` to `-1`:
+  `1/(x^2 (a + b x^3)^5)`.
+- A power of a sum of powers of x beside a symbolic power of a binomial, multiplied out into binomial differentials:
+  `(a + b x^n)^3 (c + d x^n)^(-4 - 1/n)`.
+- A difference of radicands that is one power of x, written as one: `x^4/(sqrt(a + b x) + sqrt(a + c x))^3`.
+- A rational function beside an odd root of a linear past the square, split first where a linear below meets the root
+  at an irrational value: `1/(x^3 (1 - x^2)^(1/3) (3 + x^2))`.
+- An odd integrand beside a high power of a quadratic is left to `u = x^2`, and a quartic in one shifted linear to
+  `u = d + k x`, where the partial fractions spent the budget or wrote pages.
+
+Answers the unreleased master gave change length: `x^2/((1 - a x)^16 (1 + a x)^11)` was 16,292 characters there,
+after twenty seconds, and is 1,380 in a tenth of a second; `1/((a + b/x)^(5/2) (c + d/x)^2)` was 16,675 and is 4,667.
+One is longer: `1/(x (1 - x^2)^(1/3) (3 + x^2))`, 342 characters there and 495 now.
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(a - c*x^4)/((a*d + a*k*x^2 + c*d*x^4)*sqrt(a + b*x^2 + c*x^4))".ToEntity().Integrate("x")` | `integral(...)` | 1,949 characters |
+| `"(g + h*x)*sqrt(a + b*x + c*x^2)/(a*d + b*d*x + c*d*x^2)^(3/2)".ToEntity().Integrate("x")` | `integral(...)` | 4,376 characters |
+| `"x^2/((1 - a*x)^16*(1 + a*x)^11)".ToEntity().Integrate("x")` | `integral(...)` | 1,380 characters |
+| `"1/((a + b/x)^(5/2)*(c + d/x)^2)".ToEntity().Integrate("x")` | `integral(...)` | 4,667 characters |
+| `"(c*x)^(-1 - 2*n - n*p)*(a + b*x^n)^p".ToEntity().Integrate("x")` | `integral(...)` | 255 characters |
+| `"1/(x^2*(a + b*x^3)^5)".ToEntity().Integrate("x")` | `integral(...)` | 741 characters |
+| `"(d + k*x)^4/(a + b*(d + k*x)^2 + c*(d + k*x)^4)".ToEntity().Integrate("x")` | no answer in twenty seconds | 452 characters |
+| `"1/(x^5*(a + b*x^2)^10)".ToEntity().Integrate("x")` | `integral(...)` | 652 characters, where the unreleased master gave 67,828 |
+| `"1/(x*(1 - x^2)^(1/3)*(3 + x^2))".ToEntity().Integrate("x")` | `integral(...)` | 503 characters, where the unreleased master gave 350 |
+
 ### A polynomial with symbols in it that is a binomial or an even quartic in a shifted variable is integrated in it
 
 **Answers where there were none.** `1/(c^2 x^3 + 3 b c x^2 + 3 b^2 x + 3 a b)` was left unevaluated,

Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs

@@ -726,6 +726,9 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // A whole power of a linear in the inverse sine, cosine or hyperbolic cosine of
             // `c + d x^2` with `c^2 = 1`, by its recurrence, which closes: by parts it went round.
             if ((answer = IndefiniteIntegralSolver.SolveAPowerOfAnInverseFunctionOfAQuadraticByTheRecurrence(expr, x)) is { }) return answer;
+            // An algebraic function of 1/x alone with a root in it, under x = 1/u, before the linears
+            // are written once: `1/((a + b/x)^(5/2) (c + d/x)^2)` ran past the budget that way.
+            if ((answer = IndefiniteIntegralSolver.SolveAnAlgebraicFunctionOfTheReciprocalOfX(expr, x, integrateByParts)) is { }) return answer;
             // A product of powers of quotients of linears with one linear in two of them, written
             // over each linear once with the constant that takes in front. First, since it answers
             // the question asked only, and the polynomial term below writes `1/u` as `u^(-1)` and
@@ -757,6 +760,9 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // A rational function with numbers for coefficients over a denominator past the twelfth
             // degree, by Hermite's reduction before anything searches: nothing below answered one.
             if ((answer = IndefiniteIntegralSolver.SolveAHighDegreeRationalFunctionByHermite(expr, x)) is { }) return answer;
+            // And a polynomial over high whole powers of two linears with a symbol in them, under
+            // t = L1/L2, where the partial fractions ran past the budget: `x^2/((1 - a x)^16 (1 + a x)^11)`.
+            if ((answer = IndefiniteIntegralSolver.SolveAPolynomialOverHighWholePowersOfTwoLinears(expr, x)) is { }) return answer;
             // A whole power of the square of a polynomial written out, written over its root: the rules
             // for a power of a binomial read `(a + b x^2)^4` where `(a^2 + 2 a b x^2 + b^2 x^4)^2` ran
             // past the budget.
@@ -1022,6 +1028,11 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // split, since Hearn's `e^(x^2)/x + 2x e^(x^2) ln(x) + ...` is a sum whose terms are
             // not elementary apart; at the top only, and only for a tower no closed rule reads.
             if ((answer = IndefiniteIntegralSolver.SolveByRischNormanAnsatz(expr, x)) is { }) return answer;
+            // A rational function with numbers beside an odd root of a linear past the square, split over
+            // the rationals first where a linear below meets the root at an irrational value:
+            // `1/(u^2 (1 - u)^(1/3) (3 + u))`, whose substitution is over the cube root of 4, and which the
+            // cancellation below spent five seconds declining.
+            if ((answer = IndefiniteIntegralSolver.SolveARationalFunctionBesideAnIrrationalOddRootOfALinearSplitFirst(expr, x, integrateByParts)) is { }) return answer;
             // A quotient of polynomials in x and the functions of x in it, cancelled with those
             // functions as indeterminates and its denominator factored the same way:
             // `(x^2 + 2x sin(x) + sin(x)^2)/(x + sin(x))^2` is `1`, and nothing else reads it.
@@ -1553,6 +1564,20 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // A half-odd power of a constant over a polynomial of either sign, written apart with
             // its sign in front. After the substitutions, which answer some of these without it.
             if ((answer = IndefiniteIntegralSolver.SolveByWritingAHalfOddPowerOfAReciprocalWithItsSign(expr, x, integrateByParts)) is { }) return answer;
+            // A power of x beside a whole power below -1 of a binomial with a symbol in it, by the
+            // reduction that raises the power to -1: `1/(x^2 (a + b x^3)^5)`, which the partial
+            // fractions declined past the first power.
+            if ((answer = IndefiniteIntegralSolver.SolveABinomialToAWholePowerBelowMinusOneByTheRecurrence(expr, x, integrateByParts)) is { }) return answer;
+            // A rational function beside a half-odd power of an even quartic P, as a rational
+            // function of x/sqrt(P) where it is one: Rubi's pseudo-elliptic
+            // `(a - c x^4)/((a d + a e x^2 + c d x^4) sqrt(a + b x^2 + c x^4))`. Late, since it fires
+            // on nothing a rule reading the root as written answers.
+            if ((answer = IndefiniteIntegralSolver.SolveByTheVariableOverTheRootOfAnEvenQuartic(expr, x, integrateByParts)) is { }) return answer;
+            // A factor `(c x)^t` as `x^t`, `(c x)^t x^(-t)` in front, constant on each interval.
+            if ((answer = IndefiniteIntegralSolver.SolveByWritingAPowerOfAMultipleOfXAsAPowerOfX(expr, x, integrateByParts)) is { }) return answer;
+            // Two square roots as one, up to a factor whose square is a constant, in front:
+            // `sqrt(b - a/x^2)/sqrt(a - b x^2)` is that factor over x.
+            if ((answer = IndefiniteIntegralSolver.SolveByWritingTwoRootsAsOneUpToAConstantFactor(expr, x, integrateByParts)) is { }) return answer;
             // The sign of a real-valued factor is constant between its zeros, and goes in
             // front of the antiderivative of the rest. After every rule that reads the sign
             // where it stands: `cos(x) sgn(sin(x))` is `|sin(x)|` by the substitution, and

Sources/AngouriMath/Functions/Continuous/Integration/Integration.Definition.cs

@@ -0,0 +1,51 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using System;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Calculus
+{
+    /// <summary>
+    /// A power of x beside a whole power below -1 of a binomial with a symbol in it, by the reduction
+    /// that raises the power to -1, and the root of a perfect square that leaves one: Rubi's 1.2.3.2.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ABinomialToAWholePowerBelowMinusOneIntegralTest
+    {
+        [Theory]
+        [InlineData("1/(x^2*(a + b*x^3)^5)")]
+        [InlineData("1/(x^3*(a + b*x^3)^5)")]
+        [InlineData("1/(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))")]
+        [InlineData("1/(x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))")]
+        public void RaisedToTheFirstPower(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 5000, $"{text.Length} characters of answer for {integrand}");
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 0.4);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -1.2, -0.7, 0.3, 0.8, 1.3, 2.9 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                if (want.IsNaN)
+                    continue;
+                compared++;
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart)
+                            < 1e-9 * Math.Max(1, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+            Assert.True(compared >= 5, $"only {compared} points could be compared for {integrand}");
+        }
+    }
+}

Sources/Tests/UnitTests/Calculus/ABinomialToAWholePowerBelowMinusOneIntegralTest.cs

@@ -0,0 +1,53 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using System;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Calculus
+{
+    /// <summary>
+    /// A factor <c>(c x)^t</c> written as <c>x^t</c>, with <c>(c x)^t x^(-t)</c> in front, whose
+    /// derivative is zero: Rubi's 1.1.3.2 and 1.2.3.2 with a symbol for the multiple of x.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class APowerOfAMultipleOfXAsAPowerOfXIntegralTest
+    {
+        [Theory]
+        [InlineData("(c*x)^(-1 - 2*n - n*p)*(a + b*x^n)^p")]
+        [InlineData("(c*x)^(-1 - 3*n - n*p)*(a + b*x^n)^p")]
+        [InlineData("(c*x)^(-1 - 4*n - n*p)*(a + b*x^n)^p")]
+        [InlineData("(d*x)^(-1 - 2*n*(1 + p))*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^p")]
+        public void WrittenAsAPowerOfX(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 5000, $"{text.Length} characters of answer for {integrand}");
+            // n past the wrap, so that x^n at a negative x is not confined to a half-plane.
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", -0.4).Substitute("d", 0.3)
+                .Substitute("n", 2.3).Substitute("p", 0.6);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -1.2, -0.7, 0.3, 0.8, 1.3, 2.9 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                if (want.IsNaN)
+                    continue;
+                compared++;
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart)
+                            < 1e-9 * Math.Max(1, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+            Assert.True(compared >= 5, $"only {compared} points could be compared for {integrand}");
+        }
+    }
+}

Sources/Tests/UnitTests/Calculus/APowerOfAMultipleOfXAsAPowerOfXIntegralTest.cs

@@ -0,0 +1,52 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using System;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Calculus
+{
+    /// <summary>
+    /// A whole power of a binomial in <c>x^n</c> beside a power of another with a symbol in its
+    /// exponent, written out with the power kept beside each term: each term is a binomial
+    /// differential in Chebyshev's third case. Rubi's 1.1.3.3.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AProductOfSumsBesideASymbolicPowerIntegralTest
+    {
+        [Theory]
+        [InlineData("(a + b*x^n)^3*(c + d*x^n)^(-4 - 1/n)")]
+        [InlineData("(a + b*x^n)^2*(c + d*x^n)^(-4 - 1/n)")]
+        public void WrittenOutBesideThePower(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 10000, $"{text.Length} characters of answer for {integrand}");
+            // n past the wrap, so that x^n at a negative x is not confined to a half-plane.
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 0.4).Substitute("d", 0.3)
+                .Substitute("n", 2.3);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -1.2, -0.7, 0.3, 0.8, 1.3, 2.9 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                if (want.IsNaN)
+                    continue;
+                compared++;
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart)
+                            < 1e-9 * Math.Max(1, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+            Assert.True(compared >= 5, $"only {compared} points could be compared for {integrand}");
+        }
+    }
+}

Sources/Tests/UnitTests/Calculus/AProductOfSumsBesideASymbolicPowerIntegralTest.cs

@@ -0,0 +1,49 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using System;
+using AngouriMath.Extensions;
+using Xunit;
+
+namespace AngouriMath.Tests.Calculus
+{
+    /// <summary>
+    /// A power of a linear over a trinomial in its square, written in the linear rather than split
+    /// over the quartic's quadratics first: Rubi's 1.2.3.2, which ran past the budget split.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class AQuarticInOneShiftedLinearIntegralTest
+    {
+        [Theory]
+        [InlineData("(d + k*x)^4/(a + b*(d + k*x)^2 + c*(d + k*x)^4)")]
+        [InlineData("(d + k*x)^2/(a + b*(d + k*x)^2 + c*(d + k*x)^4)")]
+        public void WrittenInTheShiftedLinear(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 5000, $"{text.Length} characters of answer for {integrand}");
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 0.4).Substitute("d", 0.9).Substitute("k", 1.1);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -1.2, -0.7, 0.3, 0.8, 1.3, 2.9 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                if (want.IsNaN)
+                    continue;
+                compared++;
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart)
+                            < 1e-9 * Math.Max(1, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+            Assert.True(compared >= 5, $"only {compared} points could be compared for {integrand}");
+        }
+    }
+}