Take a root of a perfect square with a leading coefficient of no known sign

BREAKING-CHANGES.md

@@ -541,6 +541,26 @@ quadratic with numbers in it, although beside the root each piece is a shape the
 | `"(2 + 5*x + x^2)*sqrt(3 + 2*x + 5*x^2)/(1 + 4*x - 7*x^2)^3".ToEntity().Integrate("x")` | `integral(...)` | 5,242 characters |
 | `"(a + b*x + c*x^2)^(1/2)/(d + e*x + f*x^2)^2".ToEntity().Integrate("x")` | `integral(...)` | 50,758 characters, six symbols |
 
+### A power of a perfect square with a symbolic leading coefficient is a power of its root
+
+**Answers where there were none.** `sqrt(c d^2 + 2 c d k x + c k^2 x^2)` was declined: the square root of
+`c (d + k x)^2` was taken as `sqrt(c) |d + k x|` only for a leading coefficient whose sign was known. It
+holds for every nonzero one, since `(d + k x)^2` is positive, and is taken so now. Beside a power of the
+square's own linear, that linear is taken as the root, so the two powers gather into one. The reciprocal
+root, which 2.5.0 answered in 411 characters and the unreleased master declined, is 61 characters now
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sqrt(c*d^2 + 2*c*d*k*x + c*k^2*x^2)".ToEntity().Integrate("x")` | `integral(...)` | 62 characters |
+| `"(c*d^2 + 2*c*d*k*x + c*k^2*x^2)^(3/2)".ToEntity().Integrate("x")` | `integral(...)` | 62 characters |
+| `"1/sqrt(c*d^2 + 2*c*d*k*x + c*k^2*x^2)".ToEntity().Integrate("x")` | 411 characters | 61 characters |
+| `"1/(c*d^2 + 2*c*d*k*x + c*k^2*x^2)^(5/2)".ToEntity().Integrate("x")` | `integral(...)` | 74 characters |
+| `"(d + k*x)^(-1 - 2*p)*(c*d^2 + 2*c*d*k*x + c*k^2*x^2)^p".ToEntity().Integrate("x")` | `integral(...)` | 103 characters |
+| `"1/((d + k*x)*sqrt(b^2/(4*c) + b*x + c*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 1,277 characters |
+| `"sqrt(a + a*sech(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | 274 characters |
+| `"asinh(sqrt(b*x^2 - 1))^3/sqrt(b*x^2 - 1)".ToEntity().Integrate("x")` | `integral(...)` | 166 characters |
+
 ### A square of a symbolic quadratic beside two linears is answered within the budget
 
 **Answers where there were none in time.** `1/((d + k x)(f + g x)(a + b x + c x^2)^2)` was declined by

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

@@ -21417,12 +21417,17 @@ bool MayBeARootOfASquare(Entity node, Entity.Variable x)
                 // with the answer. The other two coefficients are real or symbolic, not a
                 // number off the line, so that `(x + h)^2` is not negative.
                 // A whole power needs no sign of anything: (a (w + h)^2)^n is a^n (w + h)^(2n).
+                // And one of no known sign: for a real w + h, (w + h)^2 is positive, so
+                // Log(a (w + h)^2) is Log(a) + 2 ln|w + h| for every nonzero a, and
+                // (a (w + h)^2)^r is a^r |w + h|^(2r) -- `sqrt(c (d + k x)^2)` is
+                // `sqrt(c) |d + k x|` for a negative c as for a positive one.
                 if (!whole && a.Evaled is not Number.Real { IsZero: false })
                 {
-                    if (!IsPositiveForARealParameter(a, x, out var leadingAssumed, assumed)
-                        && !IsPositiveForARealParameter((-a).InnerSimplified, x, out leadingAssumed, assumed))
+                    if (a.Evaled is Number.Complex and not Number.Real)
                         return node;
-                    assumed = leadingAssumed;
+                    if (IsPositiveForARealParameter(a, x, out var leadingAssumed, assumed)
+                        || IsPositiveForARealParameter((-a).InnerSimplified, x, out leadingAssumed, assumed))
+                        assumed = leadingAssumed;
                 }
                 var leading = a;
                 if (b.Evaled is Number.Complex and not Number.Real || c.Evaled is Number.Complex and not Number.Real
@@ -22058,6 +22063,21 @@ static Entity Exponent(Entity exponent)
             if (found is not var (square, exponent, root, leading, squareIsBelow)
                 || above.Nodes.Concat(below.Nodes).Count(node => node == square) != 1)
                 return null;
+            // A linear of the integrand that is a multiple of the root, `d + k x` beside
+            // `(c d^2 + 2 c d k x + c k^2 x^2)^p`, is the root itself, with the multiple's square
+            // taken out of the leading coefficient: its power and the square's then gather into
+            // one power of one linear, where `(x + d/k)^(2p) (d + k x)^(-1 - 2p)` were two.
+            if (root is Sumf && TreeAnalyzer.TryGetPolyLinear(root, x, out _, out var rootShift))
+                foreach (var factor in Mulf.LinearChildren(above).Concat(Mulf.LinearChildren(below)))
+                {
+                    var linear = factor is Powf(var @base, var power) && !power.ContainsNode(x) ? @base : factor;
+                    if (linear == square || !TreeAnalyzer.TryGetPolyLinear(linear, x, out var slope, out var intercept)
+                        || slope.Evaled is Number.Complex { IsZero: true } || !VanishesIdentically(intercept - slope * rootShift))
+                        continue;
+                    root = linear;
+                    leading = leading / (slope * slope);
+                    break;
+                }
             var twice = (2 * exponent).InnerSimplified;
             // The root's power above the bar either way, with a negative exponent for the square
             // below it: `x (x + h)^(-2p)` is read below the top where `x/(x + h)^(2p)` was not.

Sources/Tests/UnitTests/Calculus/APerfectSquareWithASymbolicLeadingCoefficientIntegralTest.cs

@@ -0,0 +1,63 @@
+//
+// 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 power of a perfect square whose leading coefficient is a symbol of no known sign:
+    /// <c>(c (d + k x)^2)^r</c> is <c>(c k^2)^r |x + d/k|^(2r)</c> for a negative <c>c</c> as for a
+    /// positive one, since <c>(x + d/k)^2</c> is positive. Rubi's 1.2.1.2
+    /// <c>sqrt(c d^2 + 2 c d e x + c e^2 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 APerfectSquareWithASymbolicLeadingCoefficientIntegralTest
+    {
+        private static void DifferentiatesBack(string integrand, params (string, double)[] pins)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", 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");
+            foreach (var at in new[] { -2.3, -0.7, 0.4, 1.9 })
+            {
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                var want = original.Substitute("x", at).EvalNumerical();
+                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}");
+            }
+        }
+
+        [Theory]
+        [InlineData("sqrt(c*d^2+2*c*d*k*x+c*k^2*x^2)")]
+        [InlineData("(c*d^2+2*c*d*k*x+c*k^2*x^2)^(3/2)")]
+        [InlineData("1/sqrt(c*d^2+2*c*d*k*x+c*k^2*x^2)")]
+        [InlineData("1/(c*d^2+2*c*d*k*x+c*k^2*x^2)^(5/2)")]
+        [InlineData("(d+k*x)^(-1-2*p)*(c*d^2+2*c*d*k*x+c*k^2*x^2)^p")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            DifferentiatesBack(integrand, ("c", 3), ("d", 1.25), ("k", 2), ("p", 0.3));
+            DifferentiatesBack(integrand, ("c", -2), ("d", -0.5), ("k", 1.5), ("p", -1.7));
+        }
+    }
+}