A polynomial over a power of a quadratic beside the root of another is reduced a power at a time

[Rafael-SOWNet]

Part of #718: P/(A^k sqrt(B)) with two different quadratics and k at least two, one power of A at a time down to the closed form #1583 gives for k = 1. Symbols in the coefficients are included.

The reduction. Over A^j, P is L + A R with L linear, and R goes a power down as it is. For L = g + h x, the derivative of (p + q x) sqrt(B)/A^(j - 1) is (2 q A B + (p + q x) W)/(2 A^j sqrt(B)) with W = B' A - 2 (j - 1) B A'. So L/(A^j sqrt(B)) is that derivative plus S/(A^(j - 1) sqrt(B)) wherever 2 L = 2 q A B + (p + q x) W + 2 A S. That is five linear equations, one for each power of x up to the fourth, in p, q and the three coefficients of a quadratic S. They have a solution where A and B have no root in common. This is Hermite's reduction, which is how Rubi's 1.2.1.6 takes these. Over A itself, what is left is sigma A + L: sigma over the root alone, and L over A beside it, which #1583's rule closes.

Solved by hand. The general linear solver over the symbols took 28 s on the five equations and then failed. They don't need it: the fourth power's equation gives S2 = 2 q B2 (j - 2), the third's and the second's give S1 and S0 linear in p and q, and the first's and the constant's are then two equations in p and q, solved by their determinant. Past a bound on the size of p + q x, the reduction declines rather than grind: with six symbols the cube's coefficients reach 10,700 nodes at the second step.

Where it comes from. It is a rule of its own, beside #1583's, and a piece the rational function beside a root is now split into as well. With #1584's half-angle tangent, sqrt(a + a sec(x))/(c + d sec(x))^2 becomes (1 - t^2)^(3/2)/((1 + t^2)((c + d) + (d - c) t^2)^2) in t = tan(x/2), and the square below the bar is this. Two more things stood in the way of those rows:

• Half-odd powers of a ± a sec and a ± a csc by the half-angle tangent #1584's rule asked the chain for its rational function in t, and the substitution search ahead of the rule for a rational function beside a root spent the budget on it. It now asks that rule first, and the chain only if it declines.
• The split beside the root asked for the integral of a piece that was zero with a condition on it, 0 provided not 1 - t^2 = 0, and declined on it. Such a piece is now dropped as the zero it is.

Input	Was (2.5.0)	Master	Now
(g + h*x)/((d + k*x + f*x^2)^2*sqrt(a + b*x + c*x^2))	integral(...)	declined	a linear times the root over the quadratic, and two arctangents
(7 + 13*x)/((5 + x + 2*x^2)^3*sqrt(2 + x + 3*x^2))	integral(...)	declined	answered
sqrt(a + a*sec(x))/(c + d*sec(x))^2	integral(...)	past the budget	an antiderivative in tan(x/2)

QuadraticBesideARootIntegralTest differentiates seven more integrands back with the symbols pinned.

Measured with work/intbench against master at dd728251, the commit this is cut from, 3 s a problem:

• Rubi's rows with a half-odd power of a ± a sec or a ± a csc (970 problems): 937 → 946 solved, none lost, 0 wrong. Eight of the 4.5.2.1 timeouts Half-odd powers of a ± a sec and a ± a csc by the half-angle tangent #1584 left are answered, in 69 to 466 ms: (a + a sec)^(k/2) over the square or cube of c + d sec. So is 4.5.2.3:312, which master declined after 4.7 s. Timeouts fall from 13 to 5, and the rows take 200 s instead of 328 s.
• Rubi's 1.2.1.6, 4.3.9 and 4.4.9 (217 problems): 200 → 202. One gain is this rule's, (A + B x)/((a e + b e x + b f x^2)^2 sqrt(d + e x + f x^2)), declined by master after 19 s and answered in 2.7 s. The other, x^2 (a + b x + c x^2)^(3/2)/(d - f x^2), timed out on master and took 20 s here, past the budget either way.
• The families sample (912 problems) and family 0 (1814), on a build with this branch and three others merged, against dd728251: 793 → 804 and 1752 → 1753. Re-run on each branch alone, this branch's is family 0's Welz 43, (sqrt(x) - sqrt(x^2 - 1))^2/((1 + x - x^2)^2 sqrt(x^2 - 1)), declined by master after 9.8 s and answered in 0.6 s. None of the moves is a loss.

The unit tests pass on a build with this branch and two others of mine merged (net10.0, 13,685), and every target builds.

🤖 Generated with Claude Code