An odd power of a quadratic's derivative over a power of it, in the quadratic

[Rafael-SOWNet]

Part of #718.

(b + 2c x)^9/(a + b x + c x^2)^3 is answered through the partial fractions in 76,153 characters, with a case for each sign of the discriminant. In u = a + b x + c x^2, an odd power of a multiple of the derivative is a polynomial in u over a power of it, since (b + 2c x)^2 = 4c u + b^2 - 4ac. The answer is a few powers of the quadratic.

SolveByPartialFractions now first reads K L^m/Q^n, where Q is a quadratic, L is a linear that is a multiple lambda of Q', and m is odd and at least 3. It integrates K lambda^m (4A u + B^2 - 4AC)^((m - 1)/2)/u^n in u. A symbol shared by the linear's coefficients needs nothing more: b d + 2c d x is d times the derivative, and lambda carries it. The step stands ahead of #1836's, which still answers the even powers.

integrand	2.5.0	master 492fba61	this
(b + 2c x)^9/(a + b x + c x^2)^3	past a minute	76,153 characters, 4.5 s	607 characters, 1.2 s
(b d + 2c d x)^9/(a + b x + c x^2)^3	past a minute	76,249 characters, 4.2 s	611 characters, 0.7 s

Tests: AnOddPowerOfAQuadraticsDerivativeIntegralTest, both rows, each differentiated back and compared at six points, and each under 2,000 characters.

Measured on 14,109 corpus problems, every one whose integrand is a rational function and every one with a secant or a cosecant in it, at the corpus's 5-second budget, against master 314fa885, before #1836:

	master	this
solved	13926	13927
unevaluated	98	98
wrong	0	0
past the budget	72	71

The one problem answered here and not on master is 1.2.1.2's (b d + 2c d x)^9 row, which #1836 has since answered on master too. None go the other way. On the 13,926 both answer the time is 2,113 seconds on master and 2,101 here.

	master	this
family 0, independent suites (1814)	1782	1782
family 1, 40 a file (1381)	1341	1341
families 2 to 8, sampled (2410)	2331	2331

The harness counts no answer wrong in either. The suite passes: 15,192 passed, 13 skipped, none failed. The allocation gate passes: every gated benchmark allocates what the baseline says. Merged with master 492fba61: the library builds for every target, and the partial-fraction tests pass, #1836's among them (196). The entry in BREAKING-CHANGES.md is measured on a build of 2.5.0.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura