x^2 over a three-quarter power of a quadratic binomial beside another is integrated where that is elementary

BREAKING-CHANGES.md

@@ -419,6 +419,20 @@ leading coefficient once expanded, and declined: it is read expanded and bare no
 | `"sin(x)/(a+b*sin(x))^3".Integrate("x")`, Rubi's 4.1.2.1 row 242 | `integral(...)` | the antiderivative |
 | `"4*x*(1+x^2)/(a*x^2+2*b*x+a)^3".Integrate("x")` | `integral(...)` | the antiderivative |
 
+### `x^2` over a three-quarter power of a quadratic binomial beside another is integrated where that is elementary
+
+**Answers where there were none.** `x^2/((A + B x^2)^(3/4) (C + D x^2))` at `B C - 2 A D = 0`, Rubi's
+1.1.2.4, is the difference of the two functions whose sum answers `1/((A + B x^2)^(1/4) (C + D x^2))`
+at that ratio (the entry below), with another constant, and was declined. It is answered by the
+signs of `A` and `B` now, each form checked at points in its sign case
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"x^2/((a - b*x^2)^(3/4)*(2*a - b*x^2))".ToEntity().Integrate("x")` | `integral(...)` | an arctangent and an inverse hyperbolic tangent, by the signs of `a` and `b` |
+| `"x^2/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4))".ToEntity().Integrate("x")` | `integral(...)` | `(arctan(u) - artanh(u))/(3 sqrt(6))`, `u = sqrt(3) x/(sqrt(2) (-1 + 3 x^2)^(1/4))` |
+| `"x^2/((2 - 3*x^2)^(3/4)*(4 - 3*x^2))".ToEntity().Integrate("x")` | `integral(...)` | an arctangent and an inverse hyperbolic tangent |
+
 ### A cube or fourth root of a quadratic binomial beside another is integrated where that is elementary
 
 **Answers where there were none.** `1/((A + B x^2)^(1/3) (C + D x^2))` is an elliptic integral

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

@@ -4372,22 +4372,45 @@ private static bool TryReadAsABinomialIn(Entity expr, Entity.Variable x, out Ent
         /// it holds.
         /// https://github.com/asc-community/AngouriMath/issues/718
         /// </para>
+        /// <para>
+        /// <c>x^2/((A + B x^2)^(3/4) (C + D x^2))</c> at the same ratio is the same two functions'
+        /// difference where the quarter power is their sum, with another constant -- Rubi's
+        /// 1.1.2.4, <c>x^2/((a - b x^2)^(3/4) (2 a - b x^2))</c>, which was declined:
+        /// <code>
+        /// A &gt; 0, B &gt; 0:  -(atan(A^(3/4) (1 + s/sqrt(A))/(x y r)) - artanh(A^(3/4) (1 - s/sqrt(A))/(x y r)))/(A^(1/4) r^3)
+        /// A &gt; 0, B &lt; 0:   (atan(A^(3/4) (1 - s/sqrt(A))/(x y r)) - artanh(A^(3/4) (1 + s/sqrt(A))/(x y r)))/(A^(1/4) r^3)
+        /// A &lt; 0:           (atan(u) - artanh(u))/((-A)^(1/4) sqrt(2) r^3)
+        /// </code>
+        /// each times <c>B/D</c> as before, the constants solved for from the two functions'
+        /// derivatives and checked at points in every sign case.
+        /// </para>
         /// </remarks>
         internal static Entity? SolveAnEllipticLookingQuotientOfBinomials(Entity expr, Entity.Variable x)
         {
             var (numerator, denominator) = Functions.SingleQuotient.Of(expr);
-            if (numerator.ContainsNode(x))
-                return null;
+            // Or c x^2 above, beside a three-quarter power below.
+            var squareAbove = false;
             Entity constant = numerator;
+            if (numerator.ContainsNode(x))
+            {
+                if (!TreeAnalyzer.TryGetPolynomial(numerator, x, out var above) || above.Count != 1
+                    || !above.TryGetValue(EInteger.FromInt32(2), out var squared) || squared.ContainsNode(x))
+                    return null;
+                squareAbove = true;
+                constant = squared;
+            }
             Entity? radicand = null;
             var order = 0;
             Entity? other = null;
             foreach (var factor in Mulf.LinearChildren(denominator))
             {
                 if (!factor.ContainsNode(x))
                     constant = constant / factor;
-                else if (radicand is null && factor is Powf(var @base, Number.Rational power) && power.ERational.Numerator.Equals(EInteger.One)
-                    && (power.ERational.Denominator.Equals(EInteger.FromInt32(3)) || power.ERational.Denominator.Equals(EInteger.FromInt32(4))))
+                else if (radicand is null && factor is Powf(var @base, Number.Rational power)
+                    && (squareAbove
+                        ? power.ERational.Equals(ERational.Create(3, 4))
+                        : power.ERational.Numerator.Equals(EInteger.One)
+                          && (power.ERational.Denominator.Equals(EInteger.FromInt32(3)) || power.ERational.Denominator.Equals(EInteger.FromInt32(4)))))
                 {
                     radicand = @base;
                     order = power.ERational.Denominator.ToInt32Unchecked();
@@ -4427,7 +4450,34 @@ Entity Negative(Entity q) => twelfth * q / a * (MathS.Hyperbolic.Artanh(MathS.Po
                 var ratio = LowestOverTheSymbols(B / A);
                 answer = BySign(ratio, Positive(MathS.Sqrt(ratio)), Negative(MathS.Sqrt(LowestOverTheSymbols(-ratio))));
             }
-            else if (order == 4 && VanishesIdentically(B * C - 2 * A * D))
+            else if (order == 4 && squareAbove && VanishesIdentically(B * C - 2 * A * D))
+            {
+                var s = MathS.Sqrt(radicand);
+                var threeQuarters = Number.Rational.Create(3, 4);
+                var quarter = Number.Rational.Create(1, 4);
+                Entity BothPositive()
+                {
+                    var r = MathS.Sqrt(B);
+                    return -(MathS.Arctan(MathS.Pow(A, threeQuarters) * (1 + s / MathS.Sqrt(A)) / (x * y * r))
+                        - MathS.Hyperbolic.Artanh(MathS.Pow(A, threeQuarters) * (1 - s / MathS.Sqrt(A)) / (x * y * r))) / (MathS.Pow(A, quarter) * MathS.Pow(r, 3));
+                }
+                Entity SlopeNegative()
+                {
+                    var r = MathS.Sqrt(-B);
+                    return (MathS.Arctan(MathS.Pow(A, threeQuarters) * (1 - s / MathS.Sqrt(A)) / (x * y * r))
+                        - MathS.Hyperbolic.Artanh(MathS.Pow(A, threeQuarters) * (1 + s / MathS.Sqrt(A)) / (x * y * r))) / (MathS.Pow(A, quarter) * MathS.Pow(r, 3));
+                }
+                Entity ConstantNegative()
+                {
+                    var r = MathS.Sqrt(B);
+                    var u = x * r / (MathS.Pow(-A, quarter) * y * MathS.Sqrt(2));
+                    return (MathS.Arctan(u) - MathS.Hyperbolic.Artanh(u)) / (MathS.Pow(-A, quarter) * MathS.Sqrt(2) * MathS.Pow(r, 3));
+                }
+                var toTheRatio = LowestOverTheSymbols(B / D);
+                var form = BySigns(A, B, BothPositive(), SlopeNegative(), ConstantNegative());
+                answer = toTheRatio == Number.Integer.One ? form : toTheRatio * form;
+            }
+            else if (order == 4 && !squareAbove && VanishesIdentically(B * C - 2 * A * D))
             {
                 var s = MathS.Sqrt(radicand);
                 var threeQuarters = Number.Rational.Create(3, 4);

Sources/Tests/UnitTests/Calculus/EllipticLookingQuotientOfBinomialsTest.cs

@@ -76,6 +76,21 @@ public void AtTheRatioWhereItIsElementary(string integrand, double a, double b,
         public void WithNumbers(string integrand, double[] points)
             => DifferentiatesBack(integrand, points);
 
+        /// <summary>
+        /// <c>x^2</c> over the three-quarter power at the fourth root's ratio, the same two
+        /// functions' difference: for each sign of <c>b</c> with <c>a</c> positive, for <c>a</c>
+        /// negative, and with numbers. Rubi's 1.1.2.4.
+        /// </summary>
+        [Theory]
+        [InlineData("x^2/((a + b*x^2)^(3/4)*(2*a + b*x^2))", 1.3, 0.7, new[] { 0.3, 0.9, 1.7, -0.6 })]
+        [InlineData("x^2/((a + b*x^2)^(3/4)*(2*a + b*x^2))", 1.3, -0.7, new[] { 0.3, 0.9, 1.1, -0.6 })]
+        [InlineData("x^2/((a + b*x^2)^(3/4)*(2*a + b*x^2))", -1.3, 0.7, new[] { 1.5, 2.0, 2.6, -1.8 })]
+        [InlineData("x^2/((a - b*x^2)^(3/4)*(2*a - b*x^2))", 1.3, 0.7, new[] { 0.3, 0.9, 1.1, -0.6 })]
+        [InlineData("x^2/((-2 + 3*x^2)*(-1 + 3*x^2)^(3/4))", 0, 0, new[] { 0.7, 1.1, 1.9, -1.3 })]
+        [InlineData("x^2/((2 - 3*x^2)^(3/4)*(4 - 3*x^2))", 0, 0, new[] { 0.2, 0.5, 0.7, -0.4 })]
+        public void XSquaredOverTheThreeQuarterPower(string integrand, double a, double b, double[] points)
+            => DifferentiatesBack(integrand, points, ("a", a), ("b", b));
+
         /// <summary>
         /// Across 0, where the cube root's answers are written over <c>x</c>: the value is the one
         /// quadrature gives.