Leave a power of x past the second beside a high power of a binomial in x^2 to the binomial rules (#2041)

SolveARationalFunctionOverAPowerOfASymbolicQuadratic already declined an odd
integrand beside a power of x and a binomial in x^2 to the sixth power or
past it, leaving it to the substitution in x^2. An even one is the binomial
rules' x^m (a + b x^2)^p, and here it was peeled: Rubi's 1.1.2.2
1/(x^4 (a + b x^2)^10) took the power of x off four times, over ten levels of
the recurrence each, and came back in 75,607 characters with its like terms
never gathered. Past the second power of x it now declines, and the binomial
rules write 8,559; 1/(x^6 (a + b x^2)^8) goes from 33,638 to 6,063. Beside x^2
the binomial rules are shorter at four times the cost, 1/(x^2 (a + b x^2)^10)
in 8,322 characters and 0.49 GB against 15,658 and 0.10, and that is left as it
was.

Chain against master 78cd31b, before the narrowing to past x^2: pocket of
7,790 rows 7,788 -> 7,789 solved, 0 wrong; every family's sample unchanged;
the one move, 1.1.2.2:236, partial -> solved; of 13,328 rows compared for
allocation one grew past 1.3x and 300 MB, 1.1.2.2:235, the x^2 trade above,
which the narrowing returns to master's answer.

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

@@ -2277,6 +2277,19 @@ is written in closed form. A quadratic that factors over linears is left to the
 | `"(f + g*x)/((d + k*x)^2*(a + b*x + c*x^2)^3)".ToEntity().Integrate("x")` | `integral(...)` | 21,943 characters |
 | `"(A + B*x + C*x^2)/((d + k*x)^3*(a + c*x^2)^3)".ToEntity().Integrate("x")` | `integral(...)` | 25,201 characters |
 
+### A power of x beside a high power of a binomial in x^2 is left to the binomial rules
+
+**Shorter answers.** The rule above took a power of x off `1/(x^4 (a + b x^2)^10)`, Rubi's 1.1.2.2, four times, over ten
+levels of its recurrence each, and the answer's like terms were never gathered. Beside a power of x past the second
+and a binomial in `x^2` to the sixth power or past it, it now declines, as it already did where the integrand is odd,
+and the rules for `x^m (a + b x^2)^p` answer it. Beside `x^2` the two are a trade, shorter there at four times the
+cost, and it is left as it was ([#233](https://github.com/asc-community/AngouriMath/issues/233)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"1/(x^4*(a+b*x^2)^10)".ToEntity().Integrate("x")` | `integral(...)`; 75,607 characters on the unreleased master | 8,559 characters |
+| `"1/(x^6*(a+b*x^2)^8)".ToEntity().Integrate("x")` | `integral(...)`; 33,638 characters on the unreleased master | 6,063 characters |
+
 ### An odd power of a quadratic's derivative over a power of the quadratic is integrated in the quadratic
 
 **Shorter answers, and answers where there were none.** `(b + 2c x)^9/(a + b x + c x^2)^3` was answered through the

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

@@ -26204,14 +26204,19 @@ Entity TakeOut(Entity node)
             // a second and ten levels of this ran past the budget.
             if (Functions.PartialFractions.HoldsTheImaginaryUnit(quadratic) || linear is null && VanishesIdentically(a1))
                 return null;
-            // Nor beside a power of x where the integrand is odd and the power of the quadratic high:
-            // it is then x times a function of x^2, the substitution's under u = x^2, and peeling
-            // fifteen powers of x off `(b + 2 c x^2)/(x^15 (b + c x^2)^8)` here ran past the budget,
-            // where in u it is two linears. An even one, and a low power, are left here, where they
-            // were answered.
+            // Nor beside a power of x where the power of the quadratic is high. Odd, the integrand is
+            // x times a function of x^2, the substitution's under u = x^2, and peeling fifteen powers
+            // of x off `(b + 2 c x^2)/(x^15 (b + c x^2)^8)` here ran past the budget, where in u it is
+            // two linears. Even, and peeled past the second power of x, it is the binomial rules'
+            // `x^m (a + b x^2)^p`: peeled here four times over ten levels of the recurrence, Rubi's
+            // 1.1.2.2 `1/(x^4 (a + b x^2)^10)` came back in 75,607 characters, its like terms never
+            // gathered, and the binomial rules write 8,559. Beside `x^2` the two are a trade, shorter
+            // there at four times the cost, and it stays here. A low power is left here, where it was
+            // answered.
             if (linear is not null && power >= 6 && VanishesIdentically(a1) && TreeAnalyzer.TryGetPolyLinear(linear, x, out _, out var interceptOfTheLinear)
                 && VanishesIdentically(interceptOfTheLinear)
-                && top.Keys.All(degree => degree.IsEven != (linearPower % 2 == 0)))
+                && (top.Keys.All(degree => degree.IsEven != (linearPower % 2 == 0))
+                    || linearPower >= 3 && top.Keys.All(degree => degree.IsEven == (linearPower % 2 == 0))))
                 return null;
             // A quadratic that factors over linears is the partial fractions' to take apart, and
             // they do it cheaply: `(d + e x)^10/((d + e x) (a e + c d x))^4`, written out, went from

Sources/Tests/UnitTests/Calculus/APowerOfXBesideAHighPowerOfABinomialInXSquaredIntegralTest.cs

@@ -0,0 +1,47 @@
+//
+// 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 past the second beside a high power of a binomial in <c>x^2</c> is the binomial rules', even or odd:
+    /// the reduction over a power of a symbolic quadratic took the power of x off Rubi's 1.1.2.2
+    /// <c>1/(x^4 (a + b x^2)^10)</c> four times over ten levels each, and came back in 75,607 characters
+    /// with its like terms never gathered, where the binomial rules write 8,559.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class APowerOfXBesideAHighPowerOfABinomialInXSquaredIntegralTest
+    {
+        [Theory]
+        [InlineData("1/(x^4*(a+b*x^2)^10)", 12000)]
+        [InlineData("1/(x^6*(a+b*x^2)^8)", 9000)]
+        public void IsAnsweredInFewCharacters(string integrand, int characters)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < characters, $"the antiderivative of {integrand} is {text.Length} characters");
+            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());
+            foreach (var at in new[] { -1.7, -0.6, 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}");
+            }
+        }
+    }
+}