Split a biquadratic with surds in it by its roots in the square

BREAKING-CHANGES.md

@@ -396,6 +396,20 @@ in Rubi's 1.1.3.4 come out shorter as well, and those that took two seconds take
 | `"1/(x^8*(a + b*x^3)*(c + d*x^3))".ToEntity().Integrate("x")` | `integral(...)` | 1,217 characters |
 | `"1/(x^6*(a + b*x^4)*(c + d*x^4))".ToEntity().Integrate("x")` | `integral(...)` | 795 characters |
 
+### A biquadratic with surds in its coefficients is split by its roots in the square
+
+**Answers where there were none.** `(sqrt(2) - x^2)/(1 - sqrt(2) x^2 + x^4)` was declined, and so was Rubi's 1.2.2.3
+`(sqrt(2) + x^2)/(1 + sqrt(2) x^2 + x^4)`. An even polynomial over `a + b x^2 + c x^4` was split by the two roots in
+`x^2` only where a coefficient held a symbol, and numbers were left to the partial fractions, which factor over the
+rationals. It is now split so wherever a coefficient is not rational
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(-x^2+sqrt(2))/(1+x^4-x^2*sqrt(2))".ToEntity().Integrate("x")` | `integral(...)` | 199 characters |
+| `"(x^2+sqrt(2))/(1+x^4+x^2*sqrt(2))".ToEntity().Integrate("x")` | `integral(...)` | 201 characters |
+| `"1/(1+x^4-x^2*sqrt(3))".ToEntity().Integrate("x")` | `integral(...)` | 144 characters |
+
 ### A constant sum that comes to a number is folded before an integral is asked
 
 **Answers where there were none.** `1/sqrt(2 + 2 a - 2 (1 + a) + c x^4)` was declined, and so was Rubi's 1.1.1.2

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

@@ -3153,7 +3153,9 @@ over is Entity.Powf(var @base, var power) ?
             if (!TreeAnalyzer.TryGetPolynomial(denominator, x, out var belowAsWritten) || belowAsWritten.Count == 0
                 || belowAsWritten.Keys.Any(power => power.Sign < 0 || !power.CanFitInInt32())
                 || belowAsWritten.Values.Any(coefficient => coefficient.ContainsNode(x))
-                || !denominator.Vars.Any(v => v != x))
+                // A symbol, or a number that is not rational, which the partial fractions do not
+                // factor over: `(sqrt(2) - x^2)/(1 - sqrt(2) x^2 + x^4)` was declined.
+                || !denominator.Vars.Any(v => v != x) && belowAsWritten.Values.All(coefficient => coefficient.Evaled is Number.Rational))
                 return null;
             if (!TreeAnalyzer.TryGetPolynomial(numerator, x, out var aboveAsWritten) || aboveAsWritten.Count == 0
                 || aboveAsWritten.Keys.Any(power => power.Sign < 0 || !power.CanFitInInt32())

Sources/Tests/UnitTests/Calculus/ABiquadraticWithSurdsIntegralTest.cs

@@ -0,0 +1,57 @@
+//
+// 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>
+    /// An even polynomial over a biquadratic whose coefficients are numbers that are not rational,
+    /// split by its roots in <c>x^2</c> as a symbolic one is: Rubi's 1.2.2.3
+    /// <c>(sqrt(2) - x^2)/(1 - sqrt(2) x^2 + x^4)</c>. Roots in <c>x^2</c> complex, and real.
+    /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+    /// </summary>
+    [Trait("Area", "Calculus")]
+    public sealed class ABiquadraticWithSurdsIntegralTest
+    {
+        private static void DifferentiatesBack(string integrand, params (string, double)[] pins)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+
+            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("(-x^2+sqrt(2))/(1+x^4-x^2*sqrt(2))")]
+        [InlineData("(x^2+sqrt(2))/(1+x^4+x^2*sqrt(2))")]
+        [InlineData("1/(1+x^4-x^2*sqrt(3))")]
+        [InlineData("(1+x^2+x^4)/(1+x^4-x^2*sqrt(2))")]
+        [InlineData("1/(1+x^4-3*x^2*sqrt(2))")]
+        [InlineData("x^2/(1-x^4*sqrt(2)+x^2)")]
+        public void IsAnsweredAndDifferentiatesBack(string integrand)
+            => DifferentiatesBack(integrand);
+    }
+}