A quadratic's half-odd power beside a power of another quadratic is integrated

BREAKING-CHANGES.md

@@ -470,6 +470,24 @@ by the linear's coefficients, `(b d + 2c d x)^9`, which ran past a minute
 | `"(b + 2*c*x)^9/(a + b*x + c*x^2)^3".ToEntity().Integrate("x")` | past a minute; 76,153 characters on the unreleased master | 607 characters |
 | `"(b*d + 2*c*d*x)^9/(a + b*x + c*x^2)^3".ToEntity().Integrate("x")` | past a minute | 611 characters |
 
+### A quadratic's half-odd power beside a power of another quadratic is integrated
+
+**Answers where there were none.** `sqrt(3 - x + 2 x^2)/(2 + 3 x + 5 x^2)^2` was declined, and so were
+`1/((3 - x + 2 x^2)^(5/2) (2 + 3 x + 5 x^2))` and most of Rubi's 1.2.1.5 past the first powers. Above the bar the root
+was not read: `B^(n/2)` is `B^((n + 1)/2)` over `sqrt(B)`, a polynomial over the root, which the reduction over a power
+of the other quadratic answers. Below it, the split of the rational part over the two quadratics declined a repeated
+quadratic with numbers in it, although beside the root each piece is a shape the rules read
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"sqrt(3 - x + 2*x^2)/(2 + 3*x + 5*x^2)^2".ToEntity().Integrate("x")` | `integral(...)` | 1,344 characters |
+| `"(3 - x + 2*x^2)^(3/2)/(2 + 3*x + 5*x^2)^3".ToEntity().Integrate("x")` | `integral(...)` | 1,451 characters |
+| `"1/((3 - x + 2*x^2)^(5/2)*(2 + 3*x + 5*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 1,380 characters |
+| `"1/((3 - x + 2*x^2)^(5/2)*(2 + 3*x + 5*x^2)^3)".ToEntity().Integrate("x")` | `integral(...)` | 1,629 characters |
+| `"(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 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/Algebra/Polynomials/PartialFractions.cs

@@ -425,7 +425,12 @@ private static Entity AsFraction(RationalPolynomial numerator, RationalPolynomia
         /// https://github.com/asc-community/AngouriMath/issues/718
         /// </remarks>
         internal static bool TrySplitOverWrittenFactors(
-            Entity numerator, Entity denominator, Variable x, [NotNullWhen(true)] out Entity? decomposition)
+            Entity numerator, Entity denominator, Variable x, [NotNullWhen(true)] out Entity? decomposition,
+            // Whether the caller reads a polynomial over a power of a quadratic with numbers in
+            // it, as the rules beside the square root of a quadratic do: over the radicand's power
+            // it is a polynomial times a half-odd power of the radicand, and over another
+            // quadratic's a polynomial over a power of a quadratic beside the root of another.
+            bool aRepeatedQuadraticIsRead = false)
         {
             decomposition = null;
 
@@ -464,7 +469,7 @@ internal static bool TrySplitOverWrittenFactors(
                 // A repeated quadratic has a rule to land on only with a symbol somewhere,
                 // the split by residues below; over the rationals the system would hand it a
                 // cubic over the square, which no rule reads.
-                if (multiplicity > 1 && degree.ToInt32Unchecked() != 1 && !(numerator + denominator).Vars.Any(v => v != x))
+                if (multiplicity > 1 && degree.ToInt32Unchecked() != 1 && !aRepeatedQuadraticIsRead && !(numerator + denominator).Vars.Any(v => v != x))
                     return false;
                 foreach (var pair in read)
                     if (pair.Key.Sign < 0 || pair.Value.ContainsNode(x))

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

@@ -18271,6 +18271,24 @@ Dictionary<EInteger, Entity> PowerOfQ(int j)
                 else
                     return null;
             }
+            // The root above the bar: `B^(n/2)` for an odd `n` is `B^((n + 1)/2)/sqrt(B)`, a
+            // polynomial over the root. `sqrt(3 - x + 2 x^2)/(2 + 3 x + 5 x^2)^2` was declined,
+            // the reduction reading only a root below (Rubi's 1.2.1.5).
+            if (radicand is null && quadratic is not null)
+            {
+                Entity rest = Number.Integer.One;
+                foreach (var factor in Mulf.LinearChildren(numerator))
+                    if (radicand is null && factor is Powf(var @base, Number.Rational { ERational: var exponent })
+                        && exponent.Denominator.Equals(EInteger.FromInt32(2)) && exponent.Numerator.Sign > 0
+                        && exponent.Numerator.CanFitInInt32() && @base.ContainsNode(x))
+                    {
+                        radicand = @base;
+                        rest = rest * MathS.Pow(@base, (exponent.Numerator.ToInt32Unchecked() + 1) / 2);
+                    }
+                    else
+                        rest = rest * factor;
+                numerator = rest;
+            }
             if (radicand is null || quadratic is null || quadratic == radicand
                 || !TreeAnalyzer.TryGetPolyQuadratic(quadratic, x, out var a2, out var a1, out var a0)
                 || !TreeAnalyzer.TryGetPolyQuadratic(radicand, x, out var b2, out var b1, out var b0)
@@ -18613,7 +18631,8 @@ static Entity WithoutConditions(Entity coefficient)
             {
                 if (factors.Count == 1)
                     pieces.Add(properNumerator / factors[0]);
-                else if (Functions.PartialFractions.TrySplitOverWrittenFactors(properNumerator, linears, x, out var decomposition) && decomposition is not null)
+                else if (Functions.PartialFractions.TrySplitOverWrittenFactors(properNumerator, linears, x, out var decomposition, aRepeatedQuadraticIsRead: true)
+                         && decomposition is not null)
                 {
                     // The split comes back as a sum over a constant; the constant goes to each term.
                     var (terms, over) = decomposition is Divf(var splitTop, var splitBottom) && !splitBottom.ContainsNode(x)

Sources/Tests/UnitTests/Calculus/QuadraticBesideARootIntegralTest.cs

@@ -110,6 +110,22 @@ public void APolynomialPartBeside(string integrand)
         public void OverAPowerOfTheQuadratic(string integrand)
             => DifferentiatesBackPinned(integrand, new[] { 0.3, 0.7, 0.9, -0.5 }, Pins);
 
+        /// <summary>
+        /// The root above the bar, <c>B^(n/2)</c> as <c>B^((n + 1)/2)</c> over the root; and a
+        /// power of the radicand below it beside a power of the other quadratic, where the split
+        /// has a numeric repeated quadratic in it, a polynomial over its power beside the root
+        /// that the reductions read. Rubi's 1.2.1.5, declined before.
+        /// </summary>
+        [Theory]
+        [InlineData("sqrt(3 - x + 2*x^2)/(2 + 3*x + 5*x^2)^2")]
+        [InlineData("(3 - x + 2*x^2)^(3/2)/(2 + 3*x + 5*x^2)^3")]
+        [InlineData("(a + b*x + c*x^2)^(1/2)/(d + k*x + f*x^2)^2")]
+        [InlineData("1/((3 - x + 2*x^2)^(3/2)*(2 + 3*x + 5*x^2)^2)")]
+        [InlineData("1/((3 - x + 2*x^2)^(5/2)*(2 + 3*x + 5*x^2))")]
+        [InlineData("1/((3 - x + 2*x^2)^(5/2)*(1 + x^2))")]
+        public void TheRootAboveOrItsRadicandRepeated(string integrand)
+            => DifferentiatesBackPinned(integrand, new[] { 0.3, 0.7, 0.9, -0.5 }, Pins);
+
         /// <summary>
         /// Beside a half-odd power of <c>a + a sec(x)</c>, the half-angle tangent leaves a power
         /// of <c>(c + d) + (d - c) t^2</c> below the bar, which ran out Rubi's budget in 4.5.2.1.