Split a symbolic power of a product beside a symbolic power of one of its factors

BREAKING-CHANGES.md

@@ -733,6 +733,21 @@ The answers are as large as the split integrands' answers, which are piecewise i
 | `"(a - b*x^2)^(3/2)/sqrt(a^2 - b^2*x^4)".ToEntity().Integrate("x")` | `integral(...)` | 6,543 characters |
 | `"1/((a - b*x^2)^(1/2)*sqrt(a^2 - b^2*x^4))".ToEntity().Integrate("x")` | `integral(...)` | 877 characters |
 
+### A symbolic power of a product beside a symbolic power of one of its factors is split
+
+**Answers where there were none.** `(d + k x)^(-3 - 2p) (a d k + (c d^2 + a k^2) x + c d k x^2)^p` was declined:
+the split of a power of a product beside a power of a factor took rational exponents only. `(P Q)^r` is
+`K P^r Q^r` with `K` constant between the zeros of `P` and `Q` for any exponent free of x, so a symbolic one is
+split too, and the powers of `P` gather ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(d + k*x)^(-3 - 2*p)*(a*d*k + (c*d^2 + a*k^2)*x + c*d*k*x^2)^p".ToEntity().Integrate("x")` | not done in 60 seconds | 349 characters |
+| `"(d + k*x)^m/(a*d*k + (c*d^2 + a*k^2)*x + c*d*k*x^2)^m".ToEntity().Integrate("x")` | `integral(...)` | 215 characters |
+| `"(A + B*x)*(a*c + b*c*x)^m/(a^2 + 2*a*b*x + b^2*x^2)^(3/2)".ToEntity().Integrate("x")` | `integral(...)` | 476 characters |
+| `"(d + k*x)^m*(f + g*x)^3/(a*d*k + (c*d^2 + a*k^2)*x + c*d*k*x^2)^m".ToEntity().Integrate("x")` | `integral(...)` | 1,817 characters |
+| `"(1 - x^4)^n/(1 + x + x^2 + x^3)^n".ToEntity().Integrate("x")` | `integral(...)` | 147 characters |
+
 ### A secant or a cosecant beside a power of a sum of them is read as a power of the cosine or the sine
 
 **Answers where there were none.** `sec(x) sqrt(sec(x) + tan(x))` was declined, with the rest of Rubi's 4.7.7

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

@@ -19334,7 +19334,8 @@ Entity TakeOut(Entity node)
         }
 
         /// <summary>
-        /// A root of a product beside a root of one of its factors, the product split:
+        /// A root of a product beside a root of one of its factors, or a symbolic power of one
+        /// beside a symbolic power of a factor, the product split:
         /// <c>(P Q)^r</c> is <c>K P^r Q^r</c>, where <c>K = (P Q)^r/(P^r Q^r)</c> is constant wherever
         /// it is defined, for the same reason as in
         /// <see cref="SolveByTakingTheWholePowersOutOfAPowerOfAProduct"/>. The powers of <c>P</c>
@@ -19350,7 +19351,7 @@ Entity TakeOut(Entity node)
         {
             if (!Integration.AnsweringTheQuestionAsked || splittingARootOfAProduct)
                 return null;
-            var roots = new List<(Entity Node, Entity Base, Number.Rational Exponent)>();
+            var roots = new List<(Entity Node, Entity Base, Entity Exponent)>();
             Gather(expr);
             if (roots.Count < 2)
                 return null;
@@ -19393,7 +19394,12 @@ void Gather(Entity node)
                         Gather(above);
                         Gather(below);
                         break;
-                    case Powf(var @base, Number.Rational exponent) when exponent is not Number.Integer && @base.ContainsNode(x):
+                    // A symbolic exponent as well: K is exp(2 pi i n r) for a whole n constant
+                    // between the zeros of P and Q, whatever r is. Rubi's 1.2.1.2:2389,
+                    // `(d + e x)^(-3 - 2 p) ((d + e x) (a e + c d x))^p`, gathers into
+                    // `(d + e x)^(-3 - p) (a e + c d x)^p`.
+                    case Powf(var @base, var exponent) when exponent is not Number.Integer && !exponent.ContainsNode(x) && @base.ContainsNode(x)
+                        && (exponent is Number.Rational || exponent.Vars.Any()):
                         roots.Add((node, @base, exponent));
                         break;
                 }

Sources/Tests/UnitTests/Calculus/ASymbolicPowerOfAProductBesideAPowerOfAFactorIntegralTest.cs

@@ -0,0 +1,61 @@
+//
+// 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 symbolic power of a product of two linears beside a symbolic power of one of them:
+    /// <c>(P Q)^p</c> is a constant times <c>P^p Q^p</c>, and the powers of <c>P</c> gather.
+    /// Rubi's 1.2.1.2 <c>(d + e x)^(-3 - 2 p) (a d e + (c d^2 + a e^2) x + c d e x^2)^p</c>, written
+    /// here with <c>k</c> for <c>e</c>, which is Euler's number.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ASymbolicPowerOfAProductBesideAPowerOfAFactorIntegralTest
+    {
+        private static void DifferentiatesBack(string integrand, params (string, double)[] pins)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+
+            // Pinned before it is differentiated, with exponents whose denominators are even, so
+            // that every power of a negative base is read on the principal branch
+            // (https://github.com/asc-community/AngouriMath/issues/204).
+            var pinned = integral.Substitute("C", 0);
+            Entity original = integrand.ToEntity();
+            foreach (var (name, value) in pins)
+            {
+                pinned = pinned.Substitute(name, value);
+                original = original.Substitute(name, value);
+            }
+            var derivative = pinned.Differentiate("x");
+            foreach (var at in new[] { -2.3, -0.7, 0.4, 1.9 })
+            {
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                var want = original.Substitute("x", at).EvalNumerical();
+                var difference = Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart);
+                var scale = Math.Max(1.0, Math.Abs((double)want.RealPart) + Math.Abs((double)want.ImaginaryPart));
+                Assert.True(difference / scale < 1e-8,
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+        }
+
+        [Theory]
+        [InlineData("(d+k*x)^(-3-2*p)*(a*d*k+(c*d^2+a*k^2)*x+c*d*k*x^2)^p")]
+        [InlineData("(d+k*x)^(-5-2*p)*(a*d*k+(c*d^2+a*k^2)*x+c*d*k*x^2)^p")]
+        [InlineData("(a*d*k+(c*d^2+a*k^2)*x+c*d*k*x^2)^p/(d+k*x)^p")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            DifferentiatesBack(integrand, ("a", 2), ("c", 1.5), ("d", 1.25), ("k", 2), ("p", 0.25));
+            DifferentiatesBack(integrand, ("a", -1), ("c", 2), ("d", -0.5), ("k", 1.5), ("p", -1.75));
+        }
+    }
+}