Split a power of a product beside a whole power of a factor, and ask it early

BREAKING-CHANGES.md

@@ -766,6 +766,24 @@ split too, and the powers of `P` gather ([#718](https://github.com/asc-community
 | `"(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 power of a product beside a whole power of one of its factors below the bar is split
+
+**Answers where there were none.** `1/((d + k x)^4 (a d k + (c d^2 + a k^2) x + c d k x^2)^(3/2))` was declined
+past any budget. The split of a root of a product beside a root of a factor now also takes a factor to a whole
+power of two or more below the bar, and a product to a whole power beside a root of a factor, so the powers of
+the factor gather. It is asked before the routes that read the product whole: `(d + k x)^(7/2)/(a d k + (c d^2
++ a k^2) x + c d k x^2)` was answered by them in 29,285 characters
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"1/((d + k*x)^4*(a*d*k + (c*d^2 + a*k^2)*x + c*d*k*x^2)^(3/2))".ToEntity().Integrate("x")` | `integral(...)` | 663 characters |
+| `"1/(sqrt(d + k*x)*(a*d*k + (c*d^2 + a*k^2)*x + c*d*k*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 1,435 characters |
+| `"(d + k*x)^(7/2)/(a*d*k + (c*d^2 + a*k^2)*x + c*d*k*x^2)".ToEntity().Integrate("x")` | `integral(...)` | 1,611 characters |
+| `"(A + B*x)*(a*c + b*c*x)^m/(a^2 + 2*a*b*x + b^2*x^2)^3".ToEntity().Integrate("x")` | `integral(...)` | 223 characters |
+| `"(f + g*x)/((d + k*x)^3*(c*d^2 - b*d*k - b*k^2*x - c*k^2*x^2)^(5/2))".ToEntity().Integrate("x")` | `integral(...)` | 1,156 characters |
+| `"(12 - 3*k^2*x^2)^(1/4)/(2 + k*x)^(3/2)".ToEntity().Integrate("x")` | `integral(...)` | 613 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

@@ -19360,19 +19360,33 @@ Entity TakeOut(Entity node)
         {
             if (!Integration.AnsweringTheQuestionAsked || splittingARootOfAProduct)
                 return null;
-            var roots = new List<(Entity Node, Entity Base, Entity Exponent)>();
-            Gather(expr);
+            var roots = new List<(Entity Node, Entity Base, Entity Exponent, bool Below)>();
+            Gather(expr, false);
             if (roots.Count < 2)
                 return null;
             foreach (var product in roots)
                 foreach (var factor in roots)
                 {
+                    // One of the two a root or a symbolic power at least: two whole powers are a
+                    // rational function, which the partial fractions take apart already. And a
+                    // factor to a whole power of two or more: beside a bare one the root was split
+                    // where it read better whole -- Welz's `(1 + x)/((x - 2) sqrt(1 + x^3))` is one
+                    // logarithm, and over `sqrt(1 + x) sqrt(1 - x + x^2)` it was three.
                     if (ReferenceEquals(product.Node, factor.Node) || product.Base == factor.Base
+                        || product.Exponent is Number.Integer && factor.Exponent is Number.Integer
+                        || factor.Exponent is Number.Integer { EInteger: var once } && once.Abs().CompareTo(EInteger.FromInt32(2)) < 0
+                        // A whole power beside a root only where the split gathers powers below
+                        // the bar, and of a sum: above it, `(a + b x)^3 sqrt(c x^2 - a^2 c/b^2)` went
+                        // from 3,540 characters to 33,316, and `x^(5/2)/(b x^2 + c x^4)^3`, a power
+                        // of x, from 1.7 seconds to 7.9.
+                        || factor.Exponent is Number.Integer && !factor.Below
+                        || product.Exponent is Number.Integer && factor.Base is not (Sumf or Minusf)
                         || ExactlyDivided(product.Base, factor.Base, x) is not { } cofactor || !cofactor.ContainsNode(x))
                         continue;
                     var multiple = Variable.CreateUnique(expr, "k_split");
                     var written = MathS.Pow(factor.Base, product.Exponent) * MathS.Pow(cofactor, product.Exponent);
-                    var constant = product.Node / written;
+                    // A whole power of a product is the product of the powers, with no constant.
+                    Entity constant = product.Exponent is Number.Integer ? Number.Integer.One : product.Node / written;
                     var rewritten = expr.Replace(node => ReferenceEquals(node, product.Node) ? multiple * written : node).InnerSimplified;
                     Entity? answer;
                     splittingARootOfAProduct = true;
@@ -19391,25 +19405,35 @@ Entity TakeOut(Entity node)
                 }
             return null;
 
-            void Gather(Entity node)
+            void Gather(Entity node, bool below)
             {
                 switch (node)
                 {
                     case Mulf(var left, var right):
-                        Gather(left);
-                        Gather(right);
+                        Gather(left, below);
+                        Gather(right, below);
                         break;
-                    case Divf(var above, var below):
-                        Gather(above);
-                        Gather(below);
+                    case Divf(var above, var under):
+                        Gather(above, below);
+                        Gather(under, !below);
                         break;
                     // 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));
+                        roots.Add((node, @base, exponent, below));
+                        break;
+                    // And a whole power, or a sum standing as a factor, beside a root: Rubi's
+                    // 1.2.1.2:2218, `1/((d + e x)^4 (a d e + (c d^2 + a e^2) x + c d e x^2)^(3/2))`, is
+                    // `(d + e x)^(-11/2) (a e + c d x)^(-3/2)` up to the constant, and :2280 has the
+                    // product to the first power below the bar beside `sqrt(d + e x)`.
+                    case Powf(var @base, Number.Integer { IsZero: false } whole) when @base is Sumf or Minusf && @base.ContainsNode(x):
+                        roots.Add((node, @base, whole, below ^ whole.EInteger.Sign < 0));
+                        break;
+                    case Sumf or Minusf when node.ContainsNode(x):
+                        roots.Add((node, node, Number.Integer.One, below));
                         break;
                 }
             }

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

@@ -666,6 +666,12 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // the question asked only, and the polynomial term below writes `1/u` as `u^(-1)` and
             // asks that a level down: `1/(e^atanh(a x) (c - c/(a x))^(3/2))` reached it nowhere.
             if ((answer = IndefiniteIntegralSolver.SolveByWritingEachLinearOnce(expr, x, integrateByParts)) is { }) return answer;
+            // A root or a power of a product beside a power of one of its factors, the product
+            // split: `(a + b x^2)^(5/2)/sqrt(a^2 - b^2 x^4)` is `K (a + b x^2)^2/sqrt(a - b x^2)`.
+            // Early, since what reads the product whole is slower and longer: Rubi's 1.2.1.2:2218,
+            // `1/((d + e x)^4 (a d e + (c d^2 + a e^2) x + c d e x^2)^(3/2))`, took 27 seconds to
+            // reach it and its split form takes a quarter of one.
+            if ((answer = IndefiniteIntegralSolver.SolveBySplittingARootOfAProductBesideARootOfAFactor(expr, x, integrateByParts)) is { }) return answer;
             // A symbolic trinomial in x^2 below the bar beside a root of a quadratic, over its roots
             // in x^2, before anything reads the trinomial whole: asked of it, the routes below ran
             // past a minute on Rubi's 1.2.2.4 `sqrt(d + e x^2)/(x^2 (a + b x^2 + c x^4))`.
@@ -1239,9 +1245,6 @@ private static Entity WithItsPiecewiseSumsCombined(Entity expr)
             // And a power of a product with a factor under a whole power, that power taken out in
             // front of a constant: `(c sin(a + b x)^3)^(1/3)` is `K sin(a + b x)`.
             if ((answer = IndefiniteIntegralSolver.SolveByTakingTheWholePowersOutOfAPowerOfAProduct(expr, x, integrateByParts)) is { }) return answer;
-            // And a root of a product beside a root of one of its factors, the product split:
-            // `(a + b x^2)^(5/2)/sqrt(a^2 - b^2 x^4)` is `K (a + b x^2)^2/sqrt(a - b x^2)`.
-            if ((answer = IndefiniteIntegralSolver.SolveBySplittingARootOfAProductBesideARootOfAFactor(expr, x, integrateByParts)) is { }) return answer;
             // Product-to-sum among the rewrites rather than before them, because a product of
             // trigonometric functions of *equal* arguments is a power and wants a different tool;
             // this only fires where the arguments differ, which is exactly what every substitution

Sources/Tests/UnitTests/Calculus/APowerOfAProductBesideAWholePowerOfAFactorIntegralTest.cs

@@ -0,0 +1,60 @@
+//
+// 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 root of a product beside a whole power of one of its factors, or the product to a whole
+    /// power beside a root of a factor: split, the powers of the factor gather. Rubi's 1.2.1.2
+    /// <c>1/((d + e x)^4 (a d e + (c d^2 + a e^2) x + c d e x^2)^(3/2))</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 APowerOfAProductBesideAWholePowerOfAFactorIntegralTest
+    {
+        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: the answer carries the constant the split takes
+            // out, whose derivative is read where its arguments are numbers.
+            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("1/((d+k*x)^4*(a*d*k+(c*d^2+a*k^2)*x+c*d*k*x^2)^(3/2))")]
+        [InlineData("1/((d+k*x)^(1/2)*(a*d*k+(c*d^2+a*k^2)*x+c*d*k*x^2))")]
+        [InlineData("(d+k*x)^(7/2)/(a*d*k+(c*d^2+a*k^2)*x+c*d*k*x^2)")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            DifferentiatesBack(integrand, ("a", 2), ("c", 1.5), ("d", 1.25), ("k", 2));
+            DifferentiatesBack(integrand, ("a", -1), ("c", 2), ("d", -0.5), ("k", 1.5));
+        }
+    }
+}