A trinomial the numerator holds is divided out of a power of it

BREAKING-CHANGES.md

@@ -325,6 +325,19 @@ by the linear's coefficients, `(b d + 2c d x)^9`, which ran past a minute
 | `"(b + 2*c*x)^9/(a + b*x + c*x^2)^3".ToEntity().Integrate("x")` | past a minute; 76,153 characters on the unreleased master | 607 characters |
 | `"(b*d + 2*c*d*x)^9/(a + b*x + c*x^2)^3".ToEntity().Integrate("x")` | past a minute | 611 characters |
 
+### A trinomial the numerator holds is divided out of a power of it
+
+**A shorter answer, where 2.5.0 had none.** `(d + g x + f x^2)(a + b x^2 + c x^4)/(a + b x^2 + c x^4)^4`, written
+multiplied out as Rubi's 1.2.2.5 writes it, was declined after a minute by 2.5.0, and is answered on the unreleased
+master in 98,237 characters. The reduction of a polynomial over a power of a trinomial in `x^n` divides the trinomial
+out of the numerator first now, wherever it divides exactly, with the trinomial's content in the symbols taken out of
+it, since it arrives made monic: the answer is that of `(d + g x + f x^2)/(a + b x^2 + c x^4)^3`, 19,037 characters
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(a*d + a*g*x + (b*d + a*f)*x^2 + b*g*x^3 + (c*d + b*f)*x^4 + c*g*x^5 + c*f*x^6)/(a + b*x^2 + c*x^4)^4".ToEntity().Integrate("x")` | `integral(...)` | 19,037 characters |
+
 ### Half-odd powers of `a + a sec` and `c + d sec` together are integrated in the half angle's sine
 
 **Answers where there were none.** `sec(e + f x) sqrt(a + a sec(e + f x))/sqrt(c + d sec(e + f x))` was declined,

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

@@ -1856,6 +1856,47 @@ Entity InU(Dictionary<EInteger, Entity> polynomial)
         /// </summary>
         private const int MaximumPowerOfABinomial = 12;
 
+        /// <summary>
+        /// <paramref name="dividend"/> divided by <paramref name="divisor"/> as polynomials in all
+        /// their symbols, each over its own bar, where the division of what is above the bars is
+        /// exact; <see langword="null"/> where it is not, or where either is not a polynomial
+        /// over a bar free of <paramref name="x"/>.
+        /// </summary>
+        /// <remarks>
+        /// The bars are there because the partial fractions make a block monic before this reads
+        /// it: `a + b x^2 + c x^4` arrives as `a/c + b/c x^2 + x^4`.
+        /// </remarks>
+        private static Entity? ExactlyDivided(Entity dividend, Entity divisor, Entity.Variable x)
+        {
+            var (above, aboveBar) = Functions.SingleQuotient.Of(Functions.SingleQuotient.Combine(dividend));
+            var (below, belowBar) = Functions.SingleQuotient.Of(Functions.SingleQuotient.Combine(divisor));
+            if (aboveBar.ContainsNode(x) || belowBar.ContainsNode(x))
+                return null;
+            var variables = above.Vars.Concat(below.Vars).Distinct().OrderBy(v => v.Name, System.StringComparer.Ordinal).ToList();
+            if (variables.Count == 0 || variables.Count > MultivariatePolynomial.MaxVariables)
+                return null;
+            var indices = new Dictionary<Variable, int>();
+            for (var i = 0; i < variables.Count; i++)
+                indices[variables[i]] = i;
+            if (MultivariatePolynomial.TryParse(above, indices) is not { IsZero: false } top
+                || MultivariatePolynomial.TryParse(below, indices) is not { IsConstant: false } bottom
+                || !indices.TryGetValue(x, out var at))
+                return null;
+            // Its content in x goes below the bar: the quotient `a/c + b/c x^2 + x^4` is combined as
+            // `c (a + b x^2 + c x^4)` over `c^2`, and the numerator holds the trinomial, not the c.
+            var rest = Enumerable.Range(0, variables.Count).Where(variable => variable != at).ToList();
+            Entity content = Number.Integer.One;
+            if (Functions.PolynomialGcd.ContentIn(bottom, at, rest, 0) is { IsConstant: false } inTheSymbols
+                && bottom.DivideExact(inTheSymbols) is { } primitive)
+            {
+                bottom = primitive;
+                content = inTheSymbols.ToEntity(variables);
+            }
+            if (top.DivideExact(bottom) is not { } quotient)
+                return null;
+            return (quotient.ToEntity(variables) * belowBar / (aboveBar * content)).InnerSimplified;
+        }
+
         /// <summary>
         /// A polynomial over a power of a trinomial in <c>x^n</c>,
         /// <c>P(x)/(a + b x^n + c x^(2n))^k</c> with <c>n &gt;= 2</c> and <c>k &gt;= 2</c>, a power at
@@ -1904,6 +1945,14 @@ Entity InU(Dictionary<EInteger, Entity> polynomial)
                 else
                     return null;
             }
+            // A numerator the trinomial divides is divided by it first: Rubi's 1.2.2.5 writes
+            // `(d + e x + f x^2)(a + b x^2 + c x^4)` multiplied out over the fourth power, and over it
+            // the answer was 99,000 characters where over the third it is 20,000.
+            while (trinomial is { } && power > 1 && ExactlyDivided(numerator, trinomial, x) is { } fewer)
+            {
+                numerator = fewer;
+                power--;
+            }
             if (trinomial is null || power > MaximumPowerOfABinomial
                 || !TreeAnalyzer.TryGetPolynomial(trinomial, x, out var below) || below.Count != 3
                 || below.Keys.Any(degree => !degree.CanFitInInt32()) || below.Values.Any(coefficient => coefficient.ContainsNode(x))

Sources/Tests/UnitTests/Calculus/ATrinomialTheNumeratorHoldsIntegralTest.cs

@@ -0,0 +1,42 @@
+//
+// 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 polynomial over a power of a trinomial in <c>x^n</c> that holds the trinomial has it
+    /// divided out first. Rubi's 1.2.2.5.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ATrinomialTheNumeratorHoldsIntegralTest
+    {
+        [Fact]
+        public void DifferentiatesBack()
+        {
+            const string integrand = "(a*d + a*g*x + (b*d + a*f)*x^2 + b*g*x^3 + (c*d + b*f)*x^4 + c*g*x^5 + c*f*x^6)/(a + b*x^2 + c*x^4)^4";
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Entity Pinned(Entity e) => e.Substitute("a", 2).Substitute("b", 3).Substitute("c", 5)
+                .Substitute("d", -1).Substitute("g", 7).Substitute("f", 1);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            foreach (var at in new[] { -1.7, -0.6, 0.45, 1.3 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart)
+                            < 1e-9 * Math.Max(1, Math.Abs((double)want.RealPart)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+        }
+    }
+}