Read the root's quadratic at a symbolic pole without the slope's condition

BREAKING-CHANGES.md

@@ -466,6 +466,24 @@ the first power: for a higher power the secant's own rules answer
 | `"cos(c + d*x)^2*(A + B*sec(c + d*x) + C*sec(c + d*x)^2)/(a + b*sec(c + d*x))".ToEntity().Integrate("x")` | `integral(...)` | 12,065 characters |
 | `"sin(x)^2/(a + b*csc(x))".ToEntity().Integrate("x")` | `integral(...)` | 898 characters |
 
+### A linear with a symbolic slope beside a root of a quadratic with no linear term is integrated
+
+**Answers where there were none.** `sqrt(a + c x^2)/(x (d + k x))` was declined where `sqrt(a + b x + c x^2)/(x (d + k x))`
+was answered. The rule for a constant over a linear beside the root works at the linear's root `p = -d/k`. Without
+a linear term, the quadratic's value there came back with `provided not k = 0` attached, and the table read no
+quadratic through that condition. The value is read without the condition now: a linear's slope is not zero. A
+linear with the imaginary unit in it is still declined there, as it was by accident before
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sqrt(a + c*x^2)/(x*(d + k*x))".ToEntity().Integrate("x")` | `integral(...)` | 1,362 characters |
+| `"sqrt(a + c*x^2)/(x^3*(d + k*x))".ToEntity().Integrate("x")` | `integral(...)` | 1,710 characters |
+| `"1/(x*(d + k*x)*sqrt(a + c*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 1,067 characters |
+| `"1/(x*(d + k*x)*(a + c*x^2)^(3/2))".ToEntity().Integrate("x")` | `integral(...)` | 1,769 characters |
+| `"1/(x^3*(d + k*x)^2*sqrt(a + c*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 2,801 characters |
+| `"sqrt(a + c/x^2)/(d + k*x)".ToEntity().Integrate("x")` | `integral(...)` | 1,373 characters |
+
 ### One linear twice is not two linear powers
 
 **Answers that had no value.** `(a c + b c x)^(-3 - 2 p) (f + g x) (a^2 + 2 a b x + b^2 x^2)^p` was answered

Sources/AngouriMath/Functions/Algebra/Polynomials/PartialFractions.cs

@@ -696,7 +696,7 @@ private static Entity ExpandedOverTheSymbols(Entity expr)
         }
 
         /// <summary>Whether a number off the real line is among the nodes of <paramref name="expr"/>.</summary>
-        private static bool HoldsTheImaginaryUnit(Entity expr)
+        internal static bool HoldsTheImaginaryUnit(Entity expr)
             => expr.Nodes.Any(node => node is Complex number && number is not Real && !number.ImaginaryPart.IsZero);
 
         /// <summary>

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

@@ -18095,12 +18095,20 @@ Dictionary<EInteger, Entity> PowerOfQ(int j)
             foreach (var coefficient in new[] { a, b, c, m, n, numerator, constant })
                 if (coefficient.Evaled is Number.Complex and not Number.Real)
                     return null;
+            // And a pole off the real line is not one: the answer is written with the sign of
+            // x - p, and nothing checks it after. `1 + i a x` beside a root, what
+            // `e^(3 i atan(a x))` is written as, has its pole at `i/a`.
+            if (Functions.PartialFractions.HoldsTheImaginaryUnit(m) || Functions.PartialFractions.HoldsTheImaginaryUnit(n))
+                return null;
 
             // (x - p) with p = -n/m, and K = numerator/(m constant).
             var p = (-n / m).InnerSimplified;
             var k = (numerator / (m * constant)).InnerSimplified;
-            var atP = (a * p * p + b * p + c).InnerSimplified;
-            var slopeAtP = (2 * a * p + b).InnerSimplified;
+            // Bare: `3 (-d/k)^2 + 2` comes back `... provided not k = 0`, the linear's own slope
+            // not vanishing, and the table read no quadratic in t through the condition --
+            // `1/((d + k x) sqrt(2 + 3 x^2))` was declined where `1/((d + x) sqrt(2 + 3 x^2))` was not.
+            var atP = Functions.PartialFractions.Bare((a * p * p + b * p + c).InnerSimplified);
+            var slopeAtP = Functions.PartialFractions.Bare((2 * a * p + b).InnerSimplified);
             var t = Variable.CreateUnique(expr, "t_recip");
             var inT = 1 / MathS.Pow(atP * MathS.Sqr(t) + slopeAtP * t + a, Number.Rational.Create(1, 2));
             if (IntegralPatterns.TryStandardIntegrals(inT, t) is not { } g)
@@ -18489,12 +18497,13 @@ static Entity WithoutConditions(Entity coefficient)
                 if (coefficient.Evaled is Number.Complex and not Number.Real)
                     return null;
             var p = (-n / m).InnerSimplified;
-            var atP = (a * p * p + b * p + c).InnerSimplified;
+            // Bare, as in the rule for a single linear: the slope's condition is not a case.
+            var atP = Functions.PartialFractions.Bare((a * p * p + b * p + c).InnerSimplified);
             // A root of the radicand leaves a linear under the root, which is not this.
             if (VanishesIdentically(atP)
                 || Functions.PartialFractions.TaylorCoefficientsAtTheRoot(numerator, p, power, x) is not { } taylor)
                 return null;
-            var slopeAtP = (2 * a * p + b).InnerSimplified;
+            var slopeAtP = Functions.PartialFractions.Bare((2 * a * p + b).InnerSimplified);
             var t = Variable.CreateUnique(expr, "t_recip");
             // P(p + 1/t) t^(k - 1) is the sum of the Taylor coefficients times t^(k - 1 - j), and
             // (x - p)^k is L^k/m^k.

Sources/Tests/UnitTests/Calculus/APoleWithASymbolicSlopeBesideARootIntegralTest.cs

@@ -0,0 +1,62 @@
+//
+// 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 constant over a linear with a symbolic slope beside the root of a quadratic with no
+    /// linear term: the quadratic's value at the pole came back with the slope's condition on it,
+    /// and the table read no quadratic through it. Rubi's 1.2.1.4
+    /// <c>sqrt(a + c x^2)/(x (d + e x))</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 APoleWithASymbolicSlopeBesideARootIntegralTest
+    {
+        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("1/((d+k*x)*sqrt(2+3*x^2))")]
+        [InlineData("sqrt(a+c*x^2)/(x*(d+k*x))")]
+        [InlineData("1/(x^2*(d+k*x)*sqrt(a+c*x^2))")]
+        [InlineData("1/(x*(d+k*x)^2*sqrt(a+c*x^2))")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+        {
+            DifferentiatesBack(integrand, ("a", 2), ("c", 1.5), ("d", 1.25), ("k", 2));
+            DifferentiatesBack(integrand, ("a", -1), ("c", 2), ("d", -0.5), ("k", 1.5));
+        }
+    }
+}