Integrate a power of x beside a binomial in an odd negative power of x on both sides of 0 (#2038)

SolveByTakingAPowerOfXOutOfAFractionalPower took x^(j p) out of a sum of
whole powers of x under a fractional power p: for sqrt(a + b/x^3)/x it wrote
x^(-3/2) sqrt(a x^3 + b), which has no value at a negative x, where
a + b/x^3 is real, and fourteen of Rubi's 1.1.3.2 x^m (a + b/x^3)^(k/2) were
answered for a positive x only. Past an odd j other than 1 and -1, where j p
is not whole, the rule now declines, and the substitution in the root answers
them on both sides: sqrt(a + b/x^3)/x is a function of u = sqrt(a + b/x^3).
Where j p is whole, x^(j p) is real on both sides and the rule is kept:
(a x^3 + b x^6)^(5/3), Rubi's 1.2.3.2, is x^5 times a power of a + b x^3.

Chain against master 2a98be4: the 14 partial answers of 1.1.3.2 are solved
on both sides, and the first version's guard, on the odd power alone, lost
1.2.3.2:8; keyed on j p it is answered again, 15 of 15 alone in the probe.

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

@@ -299,6 +299,21 @@ discriminant is zero is now one linear twice ([#233](https://github.com/asc-comm
 | `"1/((a*g+b*g*x)^3*(A+B*ln(k*(a+b*x)/(c+d*x)))^2)".ToEntity().Integrate("x")` | no answer in twenty seconds | 644 characters, two powers of t |
 | `"1/(x^2*(A+B*ln(k*x))^2)".ToEntity().Integrate("x")` | `integral(...)` | 133 characters, `Ei(-(A/B + ln(k x)))`, where before this change master gave an answer with no value at any point |
 
+### A power of x beside a binomial in an odd negative power of x is integrated on both sides of 0
+
+**Answers where there were none.** `sqrt(a + b/x^3)/x` and thirteen more of Rubi's 1.1.3.2,
+`x^m (a + b/x^3)^(k/2)`, were declined. `a + b/x^3` is real on both sides of 0, and they are answered
+there in the variable under the root, where taking `x^(-3/2)` out of it would leave an answer for a
+positive x only ([#233](https://github.com/asc-community/AngouriMath/issues/233)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sqrt(a+b/x^3)/x".ToEntity().Integrate("x")` | `integral(...)` | 455 characters |
+| `"sqrt(a+b/x^3)/x^7".ToEntity().Integrate("x")` | `integral(...)` | 101 characters |
+| `"(a+b/x^3)^(3/2)*x^2".ToEntity().Integrate("x")` | `integral(...)` | 842 characters |
+| `"x^5/sqrt(a+b/x^3)".ToEntity().Integrate("x")` | `integral(...)` | 831 characters |
+| `"x^2/(a+b/x^3)^(3/2)".ToEntity().Integrate("x")` | `integral(...)` | 911 characters |
+
 ### The last digits of an evaluated sine, cosine, tangent, arcsine or arctangent
 
 **Silent, and in the hundredth digit.** `EvalNumerical` at the default hundred digits gives the

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

@@ -31474,12 +31474,25 @@ bool MayBeARootOfASquare(Entity node, Entity.Variable x)
                     // minute of search to be declined, where as written it is five seconds. Past those
                     // nothing reads the sum as written, and the factor below answers both sides:
                     // `x/sqrt(a x + b x^4)`. An even power comes out as its sign, and a fractional one
-                    // is real for a positive x only, as the sum already was.
+                    // beside a power of x that is not whole is real for a positive x only, as the sum
+                    // already was.
                     if (lowest is Number.Integer { EInteger.IsEven: false } odd && odd.EInteger.Abs().Equals(EInteger.One)
                         && !sum.Nodes.Any(node => node is Powf(var @base, var r) && @base == x && r is not Number.Integer)
                         && (Mulf.LinearChildren(rest).Where(part => part.ContainsNode(x)).ToList() is not [Sumf or Minusf]
                             || Sumf.LinearChildren(rest).All(term => PowerOfX(term, x) is (_, Number.Integer { EInteger: var degree }) && degree.CompareTo(1) <= 0 && degree.Sign >= 0)))
                         continue;
+                    // And x to any other odd power out of a sum of whole powers under a power p that
+                    // leaves `x^(j p)` fractional: the sum is real on both sides of 0 and `x^(j p)` is
+                    // not, and the answer came out for a positive x only. The substitution answers it
+                    // on both: `sqrt(a + b/x^3)/x` is a function of `u = sqrt(a + b/x^3)`, where taking
+                    // out `x^(-3/2)` left Rubi's 1.1.3.2 with no value at a negative x where the
+                    // integrand is real. A whole `j p` is real on both sides, and the factor in front
+                    // is constant on each: `(a x^3 + b x^6)^(5/3)` is `x^5` times a power of
+                    // `a + b x^3`, Rubi's 1.2.3.2.
+                    if (lowest is Number.Integer { EInteger.IsEven: false } oddPast && !oddPast.EInteger.Abs().Equals(EInteger.One)
+                        && (oddPast * power).Evaled is not Number.Integer
+                        && !sum.Nodes.Any(node => node is Powf(var @base, var r) && @base == x && r is not Number.Integer))
+                        continue;
                     // The rest of the integrand a constant times the sum's derivative is the plain
                     // substitution's, u = the sum, which answers it as a power of u, where the sum
                     // with x taken out is a search: `(x - 3 sqrt(x))^(2/3) (2 sqrt(x) - 3)/sqrt(x)`

Sources/Tests/UnitTests/Calculus/APowerOfXOutOfABinomialInAnOddNegativePowerIntegralTest.cs

@@ -0,0 +1,76 @@
+//
+// 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>
+    /// <c>a + b/x^3</c> is real on both sides of 0, and taking <c>x^(-3/2)</c> out of its root, as
+    /// <c>sqrt(a + b/x^3) = K x^(-3/2) sqrt(a x^3 + b)</c>, left an answer with no value at a negative
+    /// x where the integrand is real: Rubi's 1.1.3.2 <c>x^m (a + b/x^3)^(k/2)</c>, fourteen of them. The
+    /// substitution under the root answers on both sides.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class APowerOfXOutOfABinomialInAnOddNegativePowerIntegralTest
+    {
+        [Theory]
+        [InlineData("sqrt(a+b/x^3)/x")]
+        [InlineData("sqrt(a+b/x^3)/x^7")]
+        [InlineData("(a+b/x^3)^(3/2)*x^2")]
+        [InlineData("(a+b/x^3)^(3/2)/x^10")]
+        [InlineData("x^5/sqrt(a+b/x^3)")]
+        [InlineData("x^2/(a+b/x^3)^(3/2)")]
+        public void DifferentiatesBackOnBothSidesOfZero(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            // a + b/x^3 is positive below the root of -b/a, about -0.81, and above 0.
+            foreach (var at in new[] { -2.3, -1.7, -1.1, 0.4, 1.3, 2.2 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                Assert.False(got.IsNaN, $"d/dx of the antiderivative of {integrand} has no value at x = {at}, where the integrand is {want}");
+                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}");
+            }
+        }
+
+        /// <summary>
+        /// And an odd power taken out where <c>j p</c> is whole, which is real on both sides: Rubi's
+        /// 1.2.3.2 <c>(a x^3 + b x^6)^(5/3)</c> is <c>x^5</c> times a power of <c>a + b x^3</c>, and a
+        /// guard on the odd power alone declined it.
+        /// </summary>
+        [Fact]
+        public void TakesAnOddPowerOutWhereItsPowerIsWhole()
+        {
+            const string integrand = "(a*x^3+b*x^6)^(5/3)";
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            // a x^3 + b x^6 is positive below the root of -a/b, about -1.23, and above 0.
+            foreach (var at in new[] { -2.3, -1.7, -1.4, 0.4, 1.3, 2.2 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                Assert.False(got.IsNaN, $"d/dx of the antiderivative of {integrand} has no value at x = {at}, where the integrand is {want}");
+                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}");
+            }
+        }
+    }
+}