A quadratic with rational roots takes Euler's third substitution at one of them

BREAKING-CHANGES.md

@@ -1058,6 +1058,21 @@ secant too now, and the integrand is `sgn(cos(x)) (1 + u^2)^(3/2)/(a + b u)^2` i
 | `"sec(x)^7/(a + b*tan(x))^3".ToEntity().Integrate("x")` | `integral(...)` | 1,253 characters |
 | `"sec(x)/(a + b*cot(x))".ToEntity().Integrate("x")` | `integral(...)` | 571 characters |
 
+### A quadratic with rational roots takes Euler's third substitution at one of them
+
+**Answers where there were none.** `1/(x + sqrt(3 - 2x - x^2))` was declined. Its leading coefficient is negative and
+its constant term not a square, so the second substitution wrote `sqrt(3)` into every coefficient and the rational
+integrator could not finish. `3 - 2x - x^2` is `(1 - x)(3 + x)`: shifted to a root, the constant term is zero, and the
+third substitution answers with rational coefficients throughout. A quadratic with a positive leading coefficient
+keeps the first substitution, which was already answering it
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"1/(x + sqrt(3 - 2*x - x^2))".ToEntity().Integrate("x")` | `integral(...)` | 389 characters |
+| `"1/(x + sqrt(3 - 2*x - x^2))^3".ToEntity().Integrate("x")` | `integral(...)` | 1,160 characters |
+| `"1/(2*x + sqrt(3 - 2*x - x^2))".ToEntity().Integrate("x")` | `integral(...)` | 253 characters |
+
 ### A root of a multiple of the radicand beside Euler's sum is the same root
 
 **Answers where there were none.** `sqrt(g Q) (d + e x + f sqrt(Q))^n` was declined, with the root of `g Q` below the

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

@@ -18870,6 +18870,21 @@ private static bool VanishesIdentically(Entity expr)
             return Functions.PartialFractions.Bare(inner.Simplify()).Evaled is Number.Complex { IsZero: true };
         }
 
+        /// <summary>
+        /// A rational root of <c>a x^2 + b x + c</c> with rational coefficients, the larger, or
+        /// null where its roots are not rational.
+        /// </summary>
+        private static Entity? RationalRootOf(Entity a, Entity b, Entity c)
+        {
+            if (a.Evaled is not Number.Rational ra || b.Evaled is not Number.Rational rb || c.Evaled is not Number.Rational rc
+                || ra.Evaled is Number.Complex { IsZero: true })
+                return null;
+            var discriminant = (rb * rb - 4 * ra * rc).InnerSimplified;
+            if (discriminant.Evaled is not Number.Rational { IsNegative: false } || MathS.Sqrt(discriminant).InnerSimplified.Evaled is not Number.Rational root)
+                return null;
+            return ((-rb + root) / (2 * ra)).InnerSimplified;
+        }
+
         /// <summary>
         /// <paramref name="radicand"/> as <c>a x^2 + b x + c</c>, read through a constant below the
         /// bar as well: Rubi writes <c>(a f^2 + e x (2 d + e x))/f^2</c> for
@@ -19079,6 +19094,21 @@ node is Powf(var @base, Number.Rational r) && byRadicand.TryGetValue(@base, out
             else if (aRational) which = 1;
             else if (cRational) which = 2;
             else if (aPositive) which = 1;
+            else if (RationalRootOf(a, b, c) is { } rationalRoot)
+            {
+                // Neither root rational and a < 0, the quadratic's own roots are: `3 - 2x - x^2`
+                // is `(1 - x)(3 + x)`, and the second substitution wrote `sqrt(3)` into every
+                // coefficient, after which `1/(x + sqrt(3 - 2x - x^2))` was declined. Shifted to a
+                // root, x = y + r, the constant term is zero and the third substitution's, with
+                // rational coefficients throughout. Where a > 0 the first substitution answers
+                // first and faster: `sqrt(12x^2 + 17x + 6)/((2 + 3x)^2 (30 + 31x - 12x^2)^2)`
+                // took 1.4 s by it and timed out by the shift.
+                var y = Variable.CreateUnique(expr, "y_euler");
+                if (SolveByEulerSubstitution(expr.Substitute(x, y + rationalRoot), y) is not { } inY)
+                    return null;
+                var writtenBack = radicand.Substitute(x, y + rationalRoot).Substitute(y, x - rationalRoot);
+                return inY.Substitute(y, x - rationalRoot).Replace(node => node == writtenBack ? radicand : node).InnerSimplified;
+            }
             else if (cPositive) which = 2;
             else if (aValue is null) which = 1;
             else if (cValue is null) which = 2;

Sources/Tests/UnitTests/Calculus/EulerSubstitutionTest.cs

@@ -119,6 +119,18 @@ private static void DifferentiatesBack(string integrand, double[] points, params
         [InlineData("sqrt(x^2 + 2*x)/x", new[] { 0.3, 0.9, 1.7, 2.6 })]
         public void TheRootAtZero(string integrand, double[] points) => DifferentiatesBack(integrand, points, ("r", 1.7));
 
+        /// <summary>
+        /// Neither <c>sqrt(a)</c> nor <c>sqrt(c)</c> rational, but the quadratic's roots: shifted
+        /// to one, the root is at zero and the third substitution has rational coefficients,
+        /// where the second wrote <c>sqrt(3)</c> into every one and was declined. Rubi's 1.3.2.
+        /// </summary>
+        [Theory]
+        [InlineData("1/(x + sqrt(3 - 2*x - x^2))", new[] { -2.5, -1.2, 0.3, 0.8 })]
+        [InlineData("1/(x + sqrt(3 - 2*x - x^2))^3", new[] { -2.5, -1.2, 0.3, 0.8 })]
+        [InlineData("x/(x + sqrt(3 - 2*x - x^2))", new[] { -2.5, -1.2, 0.3, 0.8 })]
+        [InlineData("1/(2*x + sqrt(3 - 2*x - x^2))", new[] { -2.5, -1.2, 0.3, 0.8 })]
+        public void ARationalRootOfTheQuadratic(string integrand, double[] points) => DifferentiatesBack(integrand, points);
+
         /// <summary>
         /// Reached one level down, which is why the rule is not scoped: a nested radical under
         /// <c>u = sqrt(1 + x)</c>, and the remainders of by parts against an inverse function.