A power of x shared by the numerator and every term of a sum below the bar is cancelled

[Rafael-SOWNet]

Part of #718.

A power of x shared by the numerator and every term of a sum below the bar is Rubi's 1.2.4 ("improper" trinomials), and two spellings of it were not read:

integrand	2.5.0	master 15aae3d3	this
x/(a x + b x^3 + c x^5)	declined	declined	369 characters
x/(x (a + b x^2 + c x^4))	declined	declined	369 characters
1/(x (a x + b x^3 + c x^5)^2)	declined	past 60 s	0.7 s, 10,565 characters
1/(x^3 (a x + b x^3 + c x^5)^2)	declined	19.6 s, 7,520 characters	1.0 s, 12,229 characters

What changes. Two spellings, each read as what it is:

• x/(x (a + b x^2 + c x^4)) reaches the integrator as x^0/(a + b x^2 + c x^4), and no rule reads x^0 as a polynomial. The integrand's normalization, on every call, writes a zeroth power of anything with the variable in it as one now: it differs from one only where its base is zero, a point the antiderivative passes through as it passes every removable one.
• SolveByPartialFractions takes the lowest power of x out of each sum below the bar, and wrote x (a x + b x^3 + c x^5)^2 as x x^2 (a + b x^2 + c x^4)^2; the splits after it read x and x^2 as two factors, and the search ran past a minute. The powers are gathered into one now, x^3 (a + b x^2 + c x^4)^2.

Tests: PowerOfATrinomialIntegralTest.AZerothPowerOfXIsOne and .ASquareWithAPowerOfXInEachTerm, differentiated back on both sides of 4 a c - b^2.

Measured first on the 698 problems of Rubi's 1.1.4.2, 1.1.4.3 and 1.2.4.2, at the corpus's 5-second budget, against master b1af529d (the cache's base; the change is cherry-picked there):

	master	this
solved	627	635
unevaluated	67	61
past the budget	4	2

The harness counts no answer wrong in either.

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	1320
families 2 to 8, sampled (2410)	2315	2321

The harness counts no answer wrong in either.

The 18 problems the two builds disagreed on, pocket and sample together, run again one build at a time: master answers 7, declines 9 and runs past the budget on 2; this answers 17 and declines 1.

The suite on the commit measured, 9adc1473 (the change cherry-picked onto b1af529d), passes, 14,944 tests, and so does the allocation gate: every gated benchmark allocates what the baseline says. The head here, 7dc1c0d8, merges master 15aae3d3; the calculus and corpus tests pass on it, 4,365 tests, and the library builds for netstandard2.0.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura