Write whole powers of a root as powers of its radicand before measuring where it holds

BREAKING-CHANGES.md

@@ -335,6 +335,24 @@ and the complex arcsine's real part moved two units away. A caller comparing a h
 result to a pinned string will see the last digits differ; a caller comparing to a tolerance
 will not. [#1338](https://github.com/asc-community/AngouriMath/issues/1338).
 
+### Whole powers of a root are written as powers of its radicand, and hold on both sides of 0
+
+**Answers where there were none.** Rubi's 1.1.3.2 `1/(x^(7/2) sqrt(a + b x^5))` and three like it, and 1.3.2's
+`sqrt(a x^23)/sqrt(1 + x^5)` and `sqrt(a x^13)/sqrt(1 + x^5)`, and 1.1.1.2's `1/(sqrt(x) sqrt(2 x - 3))`, were answered
+`provided x >= 0`. The integrand is real
+beyond `-(a/b)^(1/5)` as a product of two imaginary roots, and the answer holds there. But it spelled `x^5` as
+`((x^(1/2))^5)^2`, whose value at a negative x has a zero imaginary part, and the measurement of where the answer holds
+could not evaluate the outer root on its cut. Under `u = sqrt(x)` a whole power of `u` that is a multiple of the
+root's order is now a power of the radicand, which it is on every branch, and the condition is gone
+([#233](https://github.com/asc-community/AngouriMath/issues/233)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"1/(x^(7/2)*sqrt(a+b*x^5))".ToEntity().Integrate("x")` | `integral(...)` | 60 characters, where before this change master gave 92 for a positive x only |
+| `"1/(x^(7/2)*sqrt(1+x^5))".ToEntity().Integrate("x")` | `integral(...)` | 50 characters |
+| `"sqrt(a*x^23)/sqrt(1+x^5)".ToEntity().Integrate("x")` | `integral(...)` | 426 characters |
+| `"1/(sqrt(x)*sqrt(-3+2*x))".ToEntity().Integrate("x")` | `integral(...)` | 141 characters, where before this change master gave an answer for a positive x only |
+
 ### A function of an inverse sine of `sqrt(1 + b x^2)` beside its derivative keeps the sign of x
 
 **A wrong answer, and answers where there were none.** `1/(asin(sqrt(1 + b x^2)) sqrt(1 + b x^2))` was

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

@@ -15817,7 +15817,12 @@ node is Powf(var radical, Number.Rational e) && e is not Number.Integer
 
             if (Integration.ComputeIndefiniteIntegral(integrand, u, integrateByParts) is not { } result)
                 return null;
-            var back = result.Substitute(u, MathS.Pow(radicalBase, Number.Rational.Create(1, q)));
+            // A whole power of u that is a multiple of q is a whole power of the radicand on every
+            // branch, `(B^(1/q))^(k q) = B^k`, and is written so: as `((x^(1/2))^5)^2`, the `x^5` in Rubi's
+            // `1/(x^(7/2) sqrt(a + b x^5))` was real at a negative x with a zero imaginary part, the outer
+            // root on its branch cut there, and the measurement below could not read the answer where
+            // the integrand is real, and kept `provided x >= 0` on an answer that holds on both sides.
+            var back = WithWholePowersOfARootAsPowersOfItsRadicand(result.Substitute(u, MathS.Pow(radicalBase, Number.Rational.Create(1, q))), radicalBase);
             // Beside a second root the integrand can be real where this radicand is negative, and
             // the condition was the whole of its domain there: the remainder parts leaves from
             // Timofeev's `arcsec(x) sqrt(x^2 - 1)/x^4`, in `w = x^2`, is
@@ -48128,6 +48133,7 @@ private static bool HoldsAnEvenRootBesides(Entity expr, Entity radicand, Entity.
         /// </remarks>
         private static Entity ConditionedOnTheRadicand(Entity answer, Entity expr, Entity radicand, Entity.Variable x)
         {
+            answer = WithWholePowersOfARootAsPowersOfItsRadicand(answer, radicand);
             var condition = radicand >= Number.Integer.Zero;
             if (!HoldsAnEvenRootBesides(expr, radicand, x))
                 return answer.Provided(condition);
@@ -48143,6 +48149,29 @@ when TheOneOtherRadicandOfAHalfOddPower(expr, radicand, x) is { } other:
             }
         }
 
+        /// <summary>
+        /// <paramref name="answer"/> with every whole power of a root of <paramref name="radicand"/> that
+        /// is a multiple of the root's order written as a power of the radicand, which it is on every
+        /// branch: <c>(B^(1/q))^(k q) = B^k</c>, nested whole powers flattened first. Spelled
+        /// <c>((x^(1/2))^5)^2</c>, the <c>x^5</c> in the answer to Rubi's 1.1.3.2
+        /// <c>1/(x^(7/2) sqrt(a + b x^5))</c> was real at a negative x with a zero imaginary part, the
+        /// outer root on its cut there, which the interval and decimal evaluations rightly refuse, and
+        /// the measurement of where the answer holds kept <c>provided x &gt;= 0</c> on an answer that holds
+        /// on both sides; so did <c>(x^(1/2))^2</c> in 1.1.1.2's <c>1/(sqrt(x) sqrt(2 x - 3))</c>.
+        /// </summary>
+        private static Entity WithWholePowersOfARootAsPowersOfItsRadicand(Entity answer, Entity radicand)
+            => answer
+                .Replace(node => node is Powf(Powf(var inner, Number.Integer k), Number.Integer m) && IsARootOf(inner, radicand)
+                    ? MathS.Pow(inner, Number.Integer.Create(k.EInteger * m.EInteger)) : node)
+                .Replace(node => node is Powf(var inner, Number.Integer { EInteger: var n }) && IsARootOf(inner, radicand)
+                    && ((Powf)inner).Exponent is Number.Rational { ERational: var root } && n.Remainder(root.Denominator).IsZero
+                    ? MathS.Pow(radicand, Number.Integer.Create(n.Divide(root.Denominator))) : node);
+
+        /// <summary>Whether <paramref name="node"/> is <c>radicand^(1/q)</c> for a whole q past 1.</summary>
+        private static bool IsARootOf(Entity node, Entity radicand)
+            => node is Powf(var @base, Number.Rational { ERational: var root }) && @base == radicand
+               && root.Numerator.Equals(EInteger.One) && root.Denominator.CompareTo(EInteger.One) > 0;
+
         /// <summary>
         /// The one base other than <paramref name="radicand"/> under a power of
         /// <paramref name="expr"/> that is not whole, where every such power is half-odd; else null.

Sources/Tests/UnitTests/Calculus/WholePowersOfTheRootAsPowersOfTheRadicandIntegralTest.cs

@@ -0,0 +1,51 @@
+//
+// 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>
+    /// Under <c>u = sqrt(x)</c>, a whole power of u that is a multiple of the root's order written as a
+    /// power of the radicand, <c>(x^(1/2))^10 = x^5</c>: spelled <c>((x^(1/2))^5)^2</c>, the answer to Rubi's
+    /// 1.1.3.2 <c>1/(x^(7/2) sqrt(a + b x^5))</c> could not be read at a negative x, its root on the cut,
+    /// and kept <c>provided x &gt;= 0</c>, where the integrand is real beyond <c>-(a/b)^(1/5)</c> and the
+    /// answer holds there.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/233">#233</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class WholePowersOfTheRootAsPowersOfTheRadicandIntegralTest
+    {
+        [Theory]
+        [InlineData("1/(x^(7/2)*sqrt(a + b*x^5))")]
+        [InlineData("1/(x^(17/2)*sqrt(1 + x^5))")]
+        [InlineData("sqrt(a*x^13)/sqrt(1 + x^5)")]
+        [InlineData("1/(sqrt(x)*sqrt(-3 + 2*x))")]
+        public void HoldsOnBothSides(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            var text = integral.Stringize();
+            Assert.DoesNotContain("integral(", text);
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            // Beyond -(a/b)^(1/5), about -1.13, the integrand is real as a product of two imaginary roots,
+            // and `1/(sqrt(x) sqrt(2 x - 3))` is real for every negative x the same way.
+            foreach (var at in new[] { -2.3, -1.4, 0.4, 1.3, 2.2 })
+            {
+                var want = original.Substitute("x", at).EvalNumerical();
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                Assert.False(got.IsNaN, $"d/dx of the antiderivative of {integrand} has no value at x = {at}, where the integrand is {want}");
+                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}");
+            }
+        }
+    }
+}