The integrand the root's substitution leaves is cancelled as written, not simplified

BREAKING-CHANGES.md

@@ -665,6 +665,23 @@ taken apart into pieces each closed the same way
 | `"(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x + 2*x^2)^(3/2))".ToEntity().Integrate("x")` | `integral(...)` | an antiderivative |
 | `"1/((x^2 + 1)*sqrt(x^2 + x + 1))".ToEntity().Integrate("x")` | `integral(...)` | a logarithm and an arctangent |
 
+### A rational function of `x^n` beside a power of a symbolic binomial is integrated without a search
+
+**Answers where there were none.** A rational function of `x^n` beside `(c + d x^n)^(k - 1/n)` is a
+rational function of `u = x/(c + d x^n)^(1/n)`, and the integrand in `u` was simplified before it
+was integrated. With symbols in it the search for a simpler form grew with the power of the
+binomial: `1/((a + b x^2)^(5/2) (1 + x^2))` took 5 s on the unreleased master, the `7/2` power ran
+past 30 s, and both were declined on 2.5.0. What the search found was the factor the substitution
+leaves on both sides of the bar, which is cancelled as it is written now, the other factors left as
+they are written ([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"1/((a + b*x^2)^(5/2)*(1 + x^2))".ToEntity().Integrate("x")` | `integral(...)` | powers of `x/sqrt(a + b x^2)`, and an arctangent or a logarithm by the sign of `a - b`, in 0.6 s |
+| `"1/((a + b*x^2)^(7/2)*(1 + x^2))".ToEntity().Integrate("x")` | `integral(...)` | the same, in 0.7 s |
+| `"1/((a + b*x^2)^(9/2)*(1 + x^2))".ToEntity().Integrate("x")` | `integral(...)` | the same, in 1 s |
+| `"x^2/((a + b*x^2)^(7/2)*(c + d*x^2))".ToEntity().Integrate("x")` | `integral(...)` | the same by the sign of `c (a d - b c)`, in 1 s |
+
 ### A power of a multiple of a quadratic's derivative beside a power of the quadratic is a binomial
 
 **Improvement, not silent.** `(b d + 2 c d x)^m (a + b x + c x^2)^p`, with a power that is not whole,

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

@@ -4207,10 +4207,17 @@ node is Powf(var xAgain, Number.Integer power) && xAgain == x && power.EInteger.
             var uToTheN = MathS.Pow(u, n);
             var oneMinusDUToTheN = 1 - d * uToTheN;
             var binomialInU = c / oneMinusDUToTheN;
-            var integrand = Functions.SingleQuotient.Combine(
+            // The factor the substitution leaves on both sides cancelled as it is written, not simplified:
+            // `(1 - u^4)/((1 + u^4)(1 - u^4))` for Timofeev's above, which the rational integrator does
+            // not cancel, and the search for a simpler form that did grows with the power of the binomial
+            // while the symbols are in it: `1/((a + b x^2)^(7/2) (1 + x^2))` ran past 90 s on it. Each
+            // base is its polynomial in u first, which collects what the substitution writes twice --
+            // `1 - (-u^2) - u^2` is 1 under a secant's half angle -- and the factors stay as they are
+            // written, which the rules after this read better than their product expanded.
+            var integrand = WithTheFactorsWrittenOnBothSidesCancelled(u, Functions.SingleQuotient.Combine(
                 inXToTheN.Substitute(xToTheN, c * uToTheN / oneMinusDUToTheN)
                 * (k.IsZero ? Number.Integer.One : MathS.Pow(binomialInU, Number.Integer.Create(k)))
-                / oneMinusDUToTheN).Simplify();
+                / oneMinusDUToTheN).InnerSimplified);
             if (integrand is Providedf(var inner, _))
                 integrand = inner;
             if (integrand.ContainsNode(x) || integrand.Nodes.Any(node => node == MathS.NaN))
@@ -4221,6 +4228,66 @@ node is Powf(var xAgain, Number.Integer power) && xAgain == x && power.EInteger.
             return answer.Nodes.Any(node => node == MathS.NaN) ? null : answer;
         }
 
+        /// <summary>
+        /// <paramref name="quotient"/> with each base that is a sum written as its polynomial in
+        /// <paramref name="u"/>, and each factor then written alike above and below the bar
+        /// cancelled, power for power; the factors are kept as factors.
+        /// </summary>
+        private static Entity WithTheFactorsWrittenOnBothSidesCancelled(Entity.Variable u, Entity quotient)
+        {
+            var (top, bottom) = Functions.SingleQuotient.Of(quotient);
+            Entity AsItsPolynomial(Entity @base)
+            {
+                if (@base is not (Sumf or Minusf) || !@base.ContainsNode(u)
+                    || !TreeAnalyzer.TryGetPolynomial(@base, u, out var terms)
+                    || terms.Any(term => term.Key.Sign < 0 || term.Value.ContainsNode(u)))
+                    return @base;
+                Entity written = Number.Integer.Zero;
+                foreach (var term in terms.OrderBy(term => term.Key))
+                {
+                    var coefficient = term.Value.InnerSimplified;
+                    if (Functions.PartialFractions.IsZeroAsAValue(coefficient))
+                        continue;
+                    var monomial = term.Key.IsZero ? coefficient : coefficient * MathS.Pow(u, Number.Integer.Create(term.Key));
+                    written = written == Number.Integer.Zero ? monomial : written + monomial;
+                }
+                return written.InnerSimplified;
+            }
+            List<(Entity Base, EInteger Power)> Factors(Entity side)
+            {
+                var factors = new List<(Entity Base, EInteger Power)>();
+                foreach (var factor in Mulf.LinearChildren(side))
+                {
+                    var (@base, power) = factor is Powf(var raised, Number.Integer exponent) && exponent.EInteger.Sign > 0
+                        ? (raised, exponent.EInteger) : (factor, EInteger.One);
+                    @base = AsItsPolynomial(@base);
+                    var at = factors.FindIndex(written => written.Base == @base);
+                    if (at < 0)
+                        factors.Add((@base, power));
+                    else
+                        factors[at] = (@base, factors[at].Power.Add(power));
+                }
+                return factors;
+            }
+            var above = Factors(top);
+            var below = Factors(bottom);
+            for (var i = 0; i < above.Count; i++)
+            {
+                if (above[i].Base is Number)
+                    continue;
+                var j = below.FindIndex(written => written.Base == above[i].Base);
+                if (j < 0)
+                    continue;
+                var common = EInteger.Min(above[i].Power, below[j].Power);
+                above[i] = (above[i].Base, above[i].Power.Subtract(common));
+                below[j] = (below[j].Base, below[j].Power.Subtract(common));
+            }
+            static Entity Product(List<(Entity Base, EInteger Power)> factors)
+                => factors.Where(written => !written.Power.IsZero).Aggregate((Entity)Number.Integer.One,
+                    (product, written) => product * (written.Power.Equals(EInteger.One) ? written.Base : MathS.Pow(written.Base, Number.Integer.Create(written.Power))));
+            return (Product(above) / Product(below)).InnerSimplified;
+        }
+
         /// <summary>
         /// Reads <paramref name="expr"/> as <c>c + d x^n</c> with <c>c</c> and <c>d</c> free of
         /// <paramref name="x"/>.

Sources/Tests/UnitTests/Calculus/DividedByTheRootIntegralTest.cs

@@ -0,0 +1,69 @@
+//
+// 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 of <c>x^n</c> times <c>(c + d x^n)^(k - 1/n)</c>, which
+    /// <c>u = x/(c + d x^n)^(1/n)</c> makes a rational function of <c>u</c>.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    /// <remarks>
+    /// Checked by differentiating back at points where the integrand is real, with the symbols
+    /// pinned after the integral is taken.
+    /// </remarks>
+    [Trait("Area", "Calculus")]
+    public sealed class DividedByTheRootIntegralTest
+    {
+        private static void DifferentiatesBack(string integrand, double[] points, params (string, double)[] pins)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            var derivative = integral.Substitute("C", 0).Differentiate("x");
+            Entity original = integrand.ToEntity();
+            foreach (var (name, value) in pins)
+            {
+                derivative = derivative.Substitute(name, value);
+                original = original.Substitute(name, value);
+            }
+            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-9,
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+            Assert.True(compared >= 3, $"only {compared} points could be compared for {integrand}");
+        }
+
+        /// <summary>Timofeev's, which is <c>1/(1 + u^4)</c> under the substitution.</summary>
+        [Fact]
+        public void TimofeevsQuarticBesideAFourthRoot()
+            => DifferentiatesBack("1/((1 + x^4)*(2 + x^4)^(1/4))", new[] { -1.3, -0.4, 0.3, 0.9, 1.7 });
+
+        /// <summary>
+        /// A higher power of the root with symbols in it, beside another quadratic: the rational
+        /// function of <c>u</c> grows with the power, and the larger ones ran past 90 s.
+        /// </summary>
+        [Theory]
+        [InlineData("1/((a + b*x^2)^(5/2)*(1 + x^2))")]
+        [InlineData("1/((a + b*x^2)^(7/2)*(1 + x^2))")]
+        [InlineData("1/((a + b*x^2)^(9/2)*(1 + x^2))")]
+        public void AHalfOddPowerOfASymbolicQuadratic(string integrand)
+            => DifferentiatesBack(integrand, new[] { -1.3, -0.4, 0.3, 0.9, 1.7 }, ("a", 1.3), ("b", 0.7));
+    }
+}