A power of a quadratic with symbols in its outer coefficients is integrated over the monic quadratic

BREAKING-CHANGES.md

@@ -177,6 +177,23 @@ the division's; with it on, the rounding is the setting's own and stays
 | `"(-60.5)!"` at 30 digits, downcasting off | `5.86118478907722232671451280188E-81` | `2.93059239453861116335725639905E-81` |
 | `"abs(75 + 316.22776601683796i)"`, downcasting off | `325` | `325.000000000000026076735749…` |
 
+### A power of a quadratic with symbols in its outer coefficients is integrated over the monic quadratic
+
+**Answers where there were none.** `x^5/(a + b x + c x^2)^4` was declined, and so were `x^6` and `x^7` over the
+fourth power and `x^5` and `x^7` over the fifth, Rubi's 1.2.1.2, while `x^5/(2 + b x + c x^2)^4` and
+`x^5/(k + m x + x^2)^4` were answered. Where the reduction declines the quadratic as written, it is integrated as
+`1/c^n` times the integral over `(x^2 + k x + m)^n`, with `k = b/c` and `m = a/c` fresh symbols while it is, and put
+back after. Each answer is in three cases by the sign of `4 a/c - (b/c)^2`
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"x^5/(a + b*x + c*x^2)^4".ToEntity().Integrate("x")` | `integral(...)` | 3,474 characters |
+| `"x^6/(a + b*x + c*x^2)^4".ToEntity().Integrate("x")` | `integral(...)` | 4,142 characters |
+| `"x^7/(a + b*x + c*x^2)^4".ToEntity().Integrate("x")` | `integral(...)` | 4,823 characters |
+| `"x^5/(a + b*x + c*x^2)^5".ToEntity().Integrate("x")` | `integral(...)` | 4,066 characters |
+| `"x^7/(a + b*x + c*x^2)^5".ToEntity().Integrate("x")` | `integral(...)` | 5,602 characters |
+
 ### `a + i a tan` beside the tangent's differential is lowered over its conjugate
 
 **Answers where there were none.** `sqrt(a + i a tan(x))/(c + d tan(x))^(3/2)` ran past thirty seconds:

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

@@ -528,8 +528,9 @@ _ when TryReadOverSquareTimesRoot(expr, x, out var overFactor, out var overRadic
                 && repetitions.EInteger.ToInt32Checked() is var n and >= 2
                 && TreeAnalyzer.TryGetPolyQuadratic(repeated, x, out var qa, out var qb, out var qc)
                 && (!numerator.ContainsNode(x) || !TreeAnalyzer.IsZero(qa))
-                && IntegrateRationalOverPowerOfQuadratic(numerator, qa, qb, qc, repeated, n, x,
-                        withoutTheFirstPower: TheFirstPowerCancels(numerator, qa, qb, qc, repeated, n, x)) is { } repeatedAnswer
+                && (IntegrateRationalOverPowerOfQuadratic(numerator, qa, qb, qc, repeated, n, x,
+                        withoutTheFirstPower: TheFirstPowerCancels(numerator, qa, qb, qc, repeated, n, x))
+                    ?? OverThePowerOfTheMonicQuadraticInNamedCoefficients(numerator, qa, qb, qc, n, x)) is { } repeatedAnswer
                     => repeatedAnswer,
 
             // ∫ (px + q)/(bx + c) dx, the same rewrite one degree down: the quotient is the
@@ -1124,6 +1125,49 @@ private static Entity IntegrateLinearOverPowerOfQuadratic(
                 LinearOverPowerOfLinear(p, q, b, c, power, x));
         }
 
+        /// <summary>
+        /// ∫ N(x)/(a x^2 + b x + c)^n dx as <c>1/a^n</c> times the integral over
+        /// <c>(x^2 + k x + m)^n</c>, with <c>k = b/a</c> and <c>m = c/a</c> each a fresh symbol
+        /// while it is integrated, where <see cref="IntegrateRationalOverPowerOfQuadratic"/>
+        /// declined the quadratic as written.
+        /// </summary>
+        /// <remarks>
+        /// With symbols in both <c>a</c> and <c>c</c> the divisions by the quadratic leave
+        /// remainders the reduction cannot read as linear: <c>x^5/(a + b x + c x^2)^4</c> was
+        /// declined, where <c>x^5/(k + m x + x^2)^4</c> and <c>x^5/(2 + b x + c x^2)^4</c> are
+        /// answered in a few hundred milliseconds. Rubi's 1.2.1.2. A leading coefficient that is a
+        /// number as written, such as the <c>1 * 1^2</c> a substitution leaves, is not one: the
+        /// substitution that wrote <c>(b d + 2 c d x)^8/(a + b x + c x^2)^3</c> in the quadratic
+        /// asks with one, and naming its other two took the answer from 65,236 characters to 202,057.
+        /// https://github.com/asc-community/AngouriMath/issues/718
+        /// </remarks>
+        private static Entity? OverThePowerOfTheMonicQuadraticInNamedCoefficients(
+            Entity numerator, Entity a, Entity b, Entity c, int power, Entity.Variable x)
+        {
+            if (a.InnerSimplified is Entity.Number)
+                return null;
+            Entity everything = numerator + a + b + c;
+            var named = new List<(Entity.Variable Name, Entity Value)>();
+            Entity Named(Entity value)
+            {
+                var written = value.InnerSimplified;
+                if (written is Entity.Variable or Entity.Number)
+                    return written;
+                var name = Entity.Variable.CreateUnique(everything, "k_q");
+                everything += name;
+                named.Add((name, written));
+                return name;
+            }
+            var linear = Named(b / a);
+            var constant = Named(c / a);
+            var monic = MathS.Pow(x, 2) + linear * x + constant;
+            if (IntegrateRationalOverPowerOfQuadratic(numerator, Entity.Number.Integer.One, linear, constant, monic, power, x) is not { } answer)
+                return null;
+            foreach (var (name, value) in named)
+                answer = answer.Substitute(name, value);
+            return answer / MathS.Pow(a, power);
+        }
+
         /// <summary>
         /// ∫ N(x)/(a x^2 + b x + c)^n dx for a numerator of any degree, or <see langword="null"/>
         /// where the fraction is improper and this is not the rule for it.

Sources/Tests/UnitTests/Calculus/APowerOfAQuadraticMadeMonicIntegralTest.cs

@@ -0,0 +1,43 @@
+//
+// 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 quadratic with symbols in both outer coefficients is
+    /// integrated over the monic quadratic in named coefficients where the reduction declines
+    /// it as written. 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 APowerOfAQuadraticMadeMonicIntegralTest
+    {
+        [Theory]
+        [InlineData("x^5/(a + b*x + c*x^2)^4")]
+        [InlineData("x^7/(a + b*x + c*x^2)^5")]
+        public void DifferentiatesBack(string integrand)
+        {
+            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);
+            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}");
+            }
+        }
+    }
+}