Factor a sextic whose leading coefficient needs a monic scale past 2^512

BREAKING-CHANGES.md

@@ -447,6 +447,19 @@ case is most of its length ([#718](https://github.com/asc-community/AngouriMath/
 | `"x^4/((a + b*x^2 + c*x^4)*sqrt(d + k*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 28,551 characters |
 | `"1/(x^2*(a + b*x^2 + c*x^4)*sqrt(d + k*x^2))".ToEntity().Integrate("x")` | `integral(...)` | 10,194 characters |
 
+### A polynomial of degree six with a large leading coefficient is factored
+
+**Answers where there were none.** The factoring clears a leading coefficient by rescaling the variable, and declined
+when that raised it past 2^512. Over the square of `(x^2 + 6)^3 + 324 x^3` the resultant the integrator factors is a sextic
+with a 107-bit leading coefficient, which needs 2^535. Up to degree six the bound is now 2^1024; past it, it stays
+([#718](https://github.com/asc-community/AngouriMath/issues/718)).
+
+| Input | Was (2.5.0) | Now |
+|---|---|---|
+| `"x^8/(216+108*x^2+324*x^3+18*x^4+x^6)^2".ToEntity().Integrate("x")` | `integral(...)` | 381 characters, a sum over the roots of the sextic |
+| `"x^5/(216+108*x^2+324*x^3+18*x^4+x^6)^2".ToEntity().Integrate("x")` | `integral(...)` | 377 characters |
+| `"x^2/(216+108*x^2+324*x^3+18*x^4+x^6)^2".ToEntity().Integrate("x")` | `integral(...)` | 394 characters |
+
 ### A rational function past the twelfth degree is reduced by Hermite's method
 
 **Answers where there were none.** The integrator read a denominator of degree twelve at most, so a derivative over

Sources/AngouriMath/Functions/Algebra/Polynomials/PolynomialFactorization.cs

@@ -91,6 +91,18 @@ internal static class PolynomialFactorization
         [ConstantField]
         private static readonly EInteger MaxMonicScale = EInteger.FromInt32(2).Pow(512);
 
+        /// <summary>
+        /// The same, up to <see cref="MaxLowDegree"/>: a sextic of Rubi's 1.3.1 whose leading
+        /// coefficient is 107 bits needs 2^535, and is factored in a quarter of a second. Past
+        /// that degree the bound stays: <c>1/(1 + cosh(x)^5)</c> asks for a hundred and twenty
+        /// polynomials of degree eight and nine at 2^580 to 2^880, a second each and none of them
+        /// needed, and went from 7 seconds to 72 with the higher bound for every degree.
+        /// </summary>
+        [ConstantField]
+        private static readonly EInteger MaxLowDegreeMonicScale = EInteger.FromInt32(2).Pow(1024);
+
+        private const int MaxLowDegree = 6;
+
         /// <summary>
         /// <paramref name="expr"/> written as a product of powers of irreducible polynomials
         /// over <c>Q</c>, or <see langword="false"/> where it is not a polynomial in
@@ -267,7 +279,7 @@ internal static bool IsWorthFactoringToSolve(Entity expr, Variable x)
                 return FactorMonic(primitive);
 
             var degree = primitive.Degree;
-            if (lead.Abs().Pow(degree - 1).CompareTo(MaxMonicScale) > 0)
+            if (lead.Abs().Pow(degree - 1).CompareTo(degree <= MaxLowDegree ? MaxLowDegreeMonicScale : MaxMonicScale) > 0)
                 return null;
 
             var scaled = new EInteger[degree + 1];

Sources/Tests/UnitTests/Calculus/RothsteinTragerTest.cs

@@ -119,5 +119,17 @@ public void TheArctangentsAreContinuous()
         [InlineData("x^2/(x^4 + x^3 + 1)")]
         public void AResidueOfDegreeThreeIsASumOverRoots(string integrand)
             => Assert.Contains(DifferentiatesBack(integrand, TheLine).Nodes, node => node is Entity.SumOverSetf);
+
+        /// <summary>
+        /// Over a square, the resultant of what Hermite's reduction leaves has a 107-bit leading
+        /// coefficient, and clearing it rescales a sextic past 2^512, where the factoring declined.
+        /// Rubi's 1.3.1 over <c>((x^2 + 6)^3 + 324 x^3)^2</c>, whose first power was answered.
+        /// <a href="https://github.com/asc-community/AngouriMath/issues/718">#718</a>
+        /// </summary>
+        [Theory]
+        [InlineData("x^8/(216+108*x^2+324*x^3+18*x^4+x^6)^2")]
+        [InlineData("x^2/(216+108*x^2+324*x^3+18*x^4+x^6)^2")]
+        public void ALargeLeadingCoefficientOfTheResultantIsCleared(string integrand)
+            => Assert.Contains(DifferentiatesBack(integrand, TheLine).Nodes, node => node is Entity.SumOverSetf);
     }
 }