A symbol a numerator sum's coefficients share comes out in front of the partial fractions

BREAKING-CHANGES.md

@@ -193,6 +193,18 @@ of a linear over a linear, which is answered in a second
 | `"1/(sqrt(a + i*a*tan(x))*(c + d*tan(x))^(3/2))".ToEntity().Integrate("x")` | `integral(...)`; past thirty seconds on the unreleased master | 2,217 characters |
 | `"1/((a + i*a*tan(x))^(3/2)*(c + d*tan(x))^(5/2))".ToEntity().Integrate("x")` | `integral(...)`; past thirty seconds on the unreleased master | 3,333 characters |
 
+### A symbol shared by a numerator sum's coefficients is taken out before the partial fractions
+
+**Answers where there were none.** `(b d + 2c d x)^9/(a + b x + c x^2)^3` ran past a minute, where
+`(b + 2c x)^9/(a + b x + c x^2)^3` was answered in two seconds: the shared `d` stood in every coefficient the
+partial fractions divide. It is taken out in front of the integral first
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(b*d + 2*c*d*x)^9/(a + b*x + c*x^2)^3".ToEntity().Integrate("x")` | past a minute | 76,249 characters, the length the content-free one already had |
+| `"(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3".ToEntity().Integrate("x")` | past a minute | 65,236 characters |
+
 ### A symbolic quadratic with a square discriminant is split into its linears before the division
 
 **Answers where there were none.** `(d + e x)^8/(a d e + (c d^2 + a e^2) x + c d e x^2)^2` ran past a minute, with

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

@@ -701,6 +701,21 @@ Entity Back(Entity mapped)
             if (!TryReadAsQuotient(expr, out var numerator, out var denominator))
                 return null;
 
+            // A power of a sum in the numerator whose coefficients share a symbol, `(b d + 2c d x)^9`,
+            // is that symbol's power times one of a sum without it: written so, it is a power of the
+            // denominator's derivative, and the division of the improper fraction as written ran past
+            // a minute where `(b + 2c x)^9/(a + b x + c x^2)^3` is answered in two seconds.
+            // The content comes out in front of the integral, not into the numerator, where its
+            // symbols would stand in every coefficient the rules below divide.
+            if (WithTheContentOutOfEachSumFactor(numerator, x) is { } numeratorWithoutContent
+                && Mulf.LinearChildren(numeratorWithoutContent).ToList() is var parts
+                && parts.Aggregate((Entity)Number.Integer.One, (product, part) => part.ContainsNode(x) ? product : product * part) is var content
+                && parts.Aggregate((Entity)Number.Integer.One, (product, part) => part.ContainsNode(x) ? product * part : product) is var contentFree
+                && Patterns.GatherPowersOfOneBase(contentFree / denominator) is var overTheContentFree
+                && (SolveByPartialFractions(overTheContentFree, x, integrateByParts)
+                    ?? Integration.ComputeIndefiniteIntegral(overTheContentFree, x, integrateByParts)) is { } withoutTheContent)
+                return content * withoutTheContent;
+
             // A quotient with x below a bar inside it, `1/(a + b/x)`, written over one bar: every
             // rule below reads the numerator and the denominator as polynomials, and `a + b/x` is
             // not one, where `x/(a x + b)` is read at once. Once: what one bar gives has none.

Sources/Tests/UnitTests/Calculus/ContentOutOfANumeratorSumIntegralTest.cs

@@ -0,0 +1,48 @@
+//
+// 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 sum in the numerator whose coefficients share a symbol, with that symbol's power
+    /// taken out in front of the integral before the partial fractions. Rubi's 1.2.1.2.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ContentOutOfANumeratorSumIntegralTest
+    {
+        [Theory]
+        [InlineData("(b*d + 2*c*d*x)^8/(a + b*x + c*x^2)^3")]
+        public void WithTheContentInFront(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Assert.True(text.Length < 100000, $"{text.Length} characters of answer for {integrand}");
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.4).Substitute("c", 1.9).Substitute("d", 1.6).Substitute("f", 0.7).Substitute("g", 0.45);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            var compared = 0;
+            foreach (var at in new[] { -2.1, -0.4, 0.3, 0.6, 1.6, 2.5 })
+            {
+                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}");
+        }
+    }
+}