The powers of a symbolic quadratic share their arms

[Rafael-SOWNet]

Part of #1788.

The reduction that takes a power off 1/(a x^2 + b x + c)^n put its arms on 4 a c - b^2 as written, and the rule for the first power puts them on it expanded. A split hands a quadratic's powers to the integrator side by side, and the piecewise that adds their answers read 4 (-c - d) = 0 and (-4) c + (-4) d = 0 as two conditions, so the arms multiplied:

integrand	2.5.0	master a5ce1c55	this
1/(x^2 - c - d)^3 + 1/(x^2 - c - d)	9 arms, 2,290 characters	9 arms, 2,518 characters	3 arms, 846 characters
2 d^3 x^2/((x^2 - c - d)^3 (x^2 - c + d))	declined	27 arms, 18,355 characters	9 arms, 6,265 characters
x^2/(a x^2 + b x + c)^2	declined	16 arms, 5,407 characters	10 arms, 2,964 characters

Each is differentiated back at six points with the symbols pinned.

What changes. IntegrateOverPowerOfQuadratic computed the expanded discriminant only to test it for zero, and wrote its arms, its roots and its reduction on the discriminant as written. It writes all of them on the expanded one now, the spelling IntegrateRationalQuadratic already uses for the first power, which the reduction bottoms out on. A lone power's answer is a little longer, 534 characters to 616 for 1/(t^2 - c - d)^3, since the expansion is written where the product was.

This is the second of three defects in the size of #1788's answers.

Tests: RepeatedQuadraticIntegralTest.ThePowersOfAQuadraticShareTheirArms, the three rows above, each with its arms counted and differentiated back at three points under two sets of values, one on each side of the discriminant's sign.

Measured first on 1,774 problems from family 1 with a symbolic quadratic or quartic to a whole power of two or more below the bar, at the corpus's 5-second budget, against master b1af529d (the cache's base; the change is cherry-picked there as 8f8e5827):

	master	this
solved	1714	1713
unevaluated	14	14
wrong	1	1
past the budget	45	46

The one counted wrong on both is #1793, fixed since in #1794.

Measured then on the Rubi corpus against master b1af529d:

	master	this
family 0, independent suites (1814)	1775	1775
family 1, 40 a file (1381)	1316	1314
families 2 to 8, sampled (2410)	2315	2313

The harness counts no answer wrong in either.

The seven problems the two builds disagreed on, pocket and sample together, run again one build at a time at the same budget, are answered by both, each in 10 to 23 seconds with the check and the two builds within 6% of each other on every one; which of them ran past the budget in the corpus runs was the load's doing. Nothing is answered or declined differently.

The suite on the commit measured, 8f8e5827 (the change cherry-picked onto b1af529d), passes, 14,988 tests, and so does the allocation gate: every gated benchmark allocates what the baseline says. The head here, 1e4c97c9, merges master a5ce1c55; the calculus and corpus tests pass on it, 4,350 tests, and the library builds for netstandard2.0.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura