Take powers of linears apart beside a symbolic root of a quadratic

BREAKING-CHANGES.md

@@ -555,6 +555,30 @@ same answer as on the unreleased master, in 1.4 seconds rather than 30
 | `"1/((d + k*x)*(f + g*x)*(a + b*x + c*x^2)^2)".ToEntity().Integrate("x")` | `integral(...)` | 24,143 characters |
 | `"1/((d + k*x)^3*(a + b*x + c*x^2)^2)".ToEntity().Integrate("x")` | `integral(...)` | 9,593 characters |
 
+### Powers of linears beside a symbolic root of a quadratic are taken apart over the linears
+
+**Answers where there were none.** `sqrt(a + b x + c x^2)/((d + k x)(f + g x)^2)` was declined: with no quadratic
+below the bar, powers of linears were left to Euler's substitution, which writes the root of a symbolic leading
+coefficient into every coefficient and does not finish. They are taken apart over the linears now where the
+integrand has a symbol in it, each piece closed beside the root. The answers are large because each piece's cases,
+by the sign of the radicand at its pole and of `c`, multiply in the sum
+([#1874](https://github.com/asc-community/AngouriMath/issues/1874)): the fourth row is 27 cases of about 3,000
+characters each ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sqrt(a + b*x + c*x^2)/((d + k*x)*(f + g*x)^2)".ToEntity().Integrate("x")` | `integral(...)` | 12,694 characters |
+| `"sqrt(a + b*x + c*x^2)/((d + k*x)*(f + g*x)^3)".ToEntity().Integrate("x")` | `integral(...)` | 20,029 characters |
+| `"sqrt(a + b*x + c*x^2)/((d + k*x)*(f + g*x)^4)".ToEntity().Integrate("x")` | `integral(...)` | 29,203 characters |
+| `"(a + b*x + c*x^2)^(3/2)/((d + k*x)*(f + g*x)^3)".ToEntity().Integrate("x")` | `integral(...)` | 79,744 characters |
+| `"1/((d + k*x)*(f + g*x)^2*sqrt(a + b*x + c*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 11,335 characters |
+| `"1/((d + k*x)*(f + g*x)^3*sqrt(a + b*x + c*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 18,103 characters |
+| `"sqrt(a + b*x^2 + c*x^4)/(x^3*(d + k*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 5,902 characters |
+| `"(a + b*x^2 + c*x^4)^(3/2)/(x^3*(d + k*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 27,184 characters |
+| `"1/(x^3*(d + k*x^2)*sqrt(a + b*x^2 + c*x^4))".ToEntity().Integrate("x")` | `integral(...)` | 5,515 characters |
+| `"asec(a + b*x)/x^4".ToEntity().Integrate("x")` | `integral(...)` | 11,873 characters |
+| `"acsc(a + b*x)/x^4".ToEntity().Integrate("x")` | `integral(...)` | 11,864 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

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

@@ -18636,8 +18636,10 @@ static Entity WithoutConditions(Entity coefficient)
             // the linears alone the reciprocal writes a sign of each, `sgn(sqrt(1 + x) - 1)` once
             // `sqrt(x + sqrt(1 + x))/x^2` is taken back from `u = sqrt(1 + x)`, whose derivative
             // is not read where the argument is not shown real -- and Euler's substitution, after
-            // this, answers those without one.
-            if (beyondTheLinears && !aQuadratic)
+            // this, answers those without one. With a symbol in them it does not: the leading
+            // coefficient's root is in every coefficient it writes, and `sqrt(a + b x + c x^2)/(x^2 (d + e x))`,
+            // which is Rubi's 1.2.2.4 under `u = x^2`, was declined.
+            if (beyondTheLinears && !aQuadratic && !expr.Vars.Any(symbol => symbol != x))
                 return null;
             // Distinct: two factors with the same root are one repeated, which is not this.
             // Beyond the linears the split decides that, and merges linears with one root.

Sources/Tests/UnitTests/Calculus/PowersOfLinearsBesideASymbolicRootIntegralTest.cs

@@ -0,0 +1,73 @@
+//
+// 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 rational function with powers of linears below the bar, beside a half-odd power of a
+    /// quadratic with symbols in it, taken apart over the linears: Euler's substitution, which
+    /// the split left these to, writes the root of a symbolic leading coefficient into every
+    /// coefficient. Rubi's 1.2.2.4 <c>sqrt(a + b x^2 + c x^4)/(x^3 (d + e x^2))</c> is one under
+    /// <c>u = x^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 PowersOfLinearsBesideASymbolicRootIntegralTest
+    {
+        private static void DifferentiatesBack(string integrand, double[] points, params (string, double)[] pins)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Assert.DoesNotContain("NaN", integral.Stringize());
+
+            // Pinned before it is differentiated: the answer carries `sgn(x + d/k)`, whose
+            // derivative is read where its argument is a number.
+            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");
+
+            var compared = 0;
+            foreach (var at in points)
+            {
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                var want = original.Substitute("x", at).EvalNumerical();
+                if (got.IsNaN || want.IsNaN)
+                    continue;
+                compared++;
+                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}");
+            }
+            Assert.True(compared >= 3,
+                $"only {compared} of {points.Length} points were comparable for {integrand}");
+        }
+
+        [Theory]
+        [InlineData("sqrt(a+b*x+c*x^2)/(x^2*(d+k*x))")]
+        [InlineData("1/(x^2*(d+k*x)*sqrt(a+b*x+c*x^2))")]
+        [InlineData("sqrt(a+b*x^2+c*x^4)/(x^3*(d+k*x^2))")]
+        [InlineData("1/((d+k*x)*(f+g*x)^2*sqrt(a+b*x+c*x^2))")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            var points = new[] { -1.4, -0.3, 0.8, 2.25 };
+            DifferentiatesBack(integrand, points, ("a", 2), ("b", 1.0 / 3), ("c", 1.5), ("d", 1.25), ("k", 3), ("f", 7.0 / 3), ("g", -0.4));
+            DifferentiatesBack(integrand, points, ("a", -1), ("b", 3), ("c", 2), ("d", -2), ("k", 1.5), ("f", 0.5), ("g", 5.0 / 3));
+        }
+    }
+}