One linear twice is not two linear powers

BREAKING-CHANGES.md

@@ -161,6 +161,21 @@ the division's; with it on, the rounding is the setting's own and stays
 | `"(-60.5)!"` at 30 digits, downcasting off | `5.86118478907722232671451280188E-81` | `2.93059239453861116335725639905E-81` |
 | `"abs(75 + 316.22776601683796i)"`, downcasting off | `325` | `325.000000000000026076735749…` |
 
+### 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
+on the unreleased master with `(1/b a b c - a c)^(-2)` in it, which is zero to a negative power: the
+answer had no value anywhere. The square is written as a power of its root, `(x + a/b)^(2 p)`, beside
+`a c + b c x`, which is `b c (x + a/b)`, and the rule for two linear powers whose exponents sum to a whole
+number divides by the two linears' cross term, declining where it is zero -- tested as a number, which
+with symbols in it it is not. It is tested as a value now, and the integral is answered by the rules
+after it
+([#1793](https://github.com/asc-community/AngouriMath/issues/1793)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"(a*c + b*c*x)^(-3-2*p)*(f + g*x)*(a^2 + 2*a*b*x + b^2*x^2)^p".ToEntity().Integrate("x")` | `integral(...)`; an answer with no value on the unreleased master | the antiderivative |
+
 ### A power of a constant below `1e-50` is no longer simplified to zero
 
 **Answers that were wrong.** Evaluation rounds a value within `1e-50` of an integer onto it, and

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

@@ -13059,8 +13059,11 @@ static bool IsRationalInBoth(Entity e, Entity.Variable x, Entity.Variable L)
             var k = -sum.ToInt32Checked();
             if (!TreeAnalyzer.TryGetPolyLinear(first, x, out var b1, out var a1) || !TreeAnalyzer.TryGetPolyLinear(second, x, out var b2, out var a2))
                 return null;
+            // Zero as a value, not only as a number: `x + a/b` beside `a c + b c x` is one linear
+            // twice, their cross term `a/b (b c) - a c` has symbols in it and is zero, and the
+            // formula below divided by it -- an answer with no value anywhere.
             var d = (a1 * b2 - a2 * b1).InnerSimplified;
-            if (d.Evaled is Number.Complex { IsZero: true })
+            if (Functions.PartialFractions.IsZeroAsAValue(d))
                 return null;
             Entity product = Number.Integer.One;
             foreach (var (factor, _) in polynomial)

Sources/Tests/UnitTests/Calculus/TwoLinearPowersIntegralTest.cs

@@ -46,5 +46,32 @@ Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.9).Substitut
                     $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
             }
         }
+
+        /// <summary>
+        /// One linear twice is not two: `x + a/b` beside `a c + b c x` has a cross term
+        /// `a/b (b c) - a c` that is zero with symbols in it, and the formula divided by it, for an
+        /// answer with no value anywhere. Rubi's 1.2.1.3:2459.
+        /// https://github.com/asc-community/AngouriMath/issues/1793
+        /// </summary>
+        [Theory]
+        [InlineData("(a*c + b*c*x)^(-3 - 2*p)*(f + g*x)*(a^2 + 2*a*b*x + b^2*x^2)^p")]
+        [InlineData("(x + a/b)^(2*p)*(a*c + b*c*x)^(-3 - 2*p)*(f + g*x)")]
+        public void OneLinearTwiceHasAValue(string integrand)
+        {
+            var integral = integrand.ToEntity().Integrate("x");
+            Assert.DoesNotContain("integral(", integral.Stringize());
+            Entity Pinned(Entity e) => e.Substitute("a", 1.3).Substitute("b", 0.7).Substitute("c", 1.1)
+                .Substitute("f", 0.3).Substitute("g", 1.7).Substitute("p", 0.29);
+            var derivative = Pinned(integral.Substitute("C", 0)).Differentiate("x");
+            var original = Pinned(integrand.ToEntity());
+            foreach (var at in new[] { 0.3, 0.8, 1.5, 2.4 })
+            {
+                var got = derivative.Substitute("x", at).EvalNumerical();
+                var want = original.Substitute("x", at).EvalNumerical();
+                Assert.False(got.IsNaN, $"the antiderivative of {integrand} has no derivative at x = {at}: {integral}");
+                Assert.True(Math.Abs((double)(got - want).RealPart) + Math.Abs((double)(got - want).ImaginaryPart) < 1e-9 * Math.Max(1.0, Math.Abs((double)want.RealPart)),
+                    $"d/dx of the antiderivative of {integrand} is {got} at x = {at}, where the integrand is {want}");
+            }
+        }
     }
 }