A symbolic quadratic with a square discriminant, split into its linears where the numerator shares one

[Rafael-SOWNet]

Part of #718.

(d + e x)^8/(a d e + (c d^2 + a e^2) x + c d e x^2)^2 ran past a minute, with eleven more of Rubi's 1.2.1.2: a power of a linear over a power of a quadratic that is that linear times another. The quadratic's discriminant is the square of a polynomial in the symbols, (c d^2 - a e^2)^2, so it is (d + e x)(a e + c d x). Over the quadratic as written, the division of the improper fraction and a substitution search ran out the time. Over its linears the shared power cancels and the rest is answered in a second.

SolveByPartialFractions now writes such a quadratic as (2A x + B - S)(2A x + B + S)/(4A) before the division, with S from MultivariatePolynomial.TrySquareRoot (#1831). It does this only where a polynomial numerator vanishes at the root of one of the two linears, which is what the split is for. A quadratic made monic first, with its coefficients over the leading one's symbols, is written over one bar and its numerator split. The numerator's sums are spelled as the split spells its linears, so the shared power is gathered and cancelled before anything else reads the quotient. #1831's helper for a quartic in x^2 now takes either step, and its call moves above a comment it had been put in the middle of.

integrand	2.5.0	master 24351278	this
(d + e x)^8/(a d e + (c d^2 + a e^2) x + c d e x^2)^2	past a minute	past a minute	5,126 characters, 2.1 s
(a + b x)^6/(a c + (b c + a d) x + b d x^2)^2	past a minute	past a minute	3,592 characters, 0.4 s
(d + e x)^10/(a d e + (c d^2 + a e^2) x + c d e x^2)^4	past a minute	past a minute	26,054 characters, 2.2 s
csc(x)^3/(a + b tan(x)^2)^2	declined	past a minute	7,992 characters, 0.2 s

x stands for e + f x in the last row, which is what was probed.

The first version split every such quadratic. On 7.4.1's (a + b arccoth(c x)) (d + e ln(1 - c^2 x^2)), which master answers in 13 seconds, parts left 1 - c^2 x^2 with nothing to cancel against. Split, it ran past two minutes. Now the split happens only where the numerator is a polynomial sharing a linear, and that problem is answered in 12.8 seconds as before.

Tests: ASymbolicQuadraticSplitIntegralTest, the first row and (a + b x)^6/(a c + (b c + a d) x + b d x^2)^3, each differentiated back and compared at six points. The pins put c d above one. Below about 0.6, the conditions these answers carry, not (c d)^68 = 0 and the like, evaluate false: (0.54)^60 = 0 is True. Master's answers have the same conditions. That is #1376's defect, and the case is recorded there.

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 24351278:

	master	this
solved	13912	13926
unevaluated	98	98
wrong	0	0
past the budget	86	72

Fourteen problems are answered here and not on master, twelve of 1.2.1.2 and two of 4.3.7, and none the other way. On the 13,912 both answer the time is 2,126 seconds on master and 2,102 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 fourteen problems the builds disagreed on, run again one build at a time: master answers none within the budget, this all fourteen.

The suite passes: 15,190 passed, 13 skipped, none failed. The allocation gate passes: every gated benchmark allocates what the baseline says. The library builds for every target. 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