A sum that is the derivative of a product of powers is read as one

[Rafael-SOWNet]

Part of #718.

m x^(m - 1) (a + b x^n)^p + b n p x^(m + n - 1) (a + b x^n)^(p - 1) is the derivative of x^m (a + b x^n)^p, written out as a sum, and was declined:

integrand	master 15aae3d3	this
m x^(m - 1) (a + b x^n)^p + b n p x^(m + n - 1) (a + b x^n)^(p - 1)	declined	51 characters
m x^(m - 1)/sqrt(a + b x^n) - b n x^(m + n - 1)/(2 (a + b x^n)^(3/2))	declined	47 characters

What changes. SolveAsTheDerivativeOfAProductOfPowers reads a product of powers beside one bracket and tries every subset of the powers raised by one; a sum of such products never matched it, and split into its terms by linearity neither term is elementary. A sum whose terms are each a constant times powers of the same bases is written now as their common product, each base to the least of its exponents -- the one every other exceeds by a whole number or by a sum of symbols with no sign in front, x^(m - 1) beside x^(m + n - 1) -- times the bracket of what is left, and asked so. The exponents' differences are simplified: the inner simplification leaves -1 + m + n - (-1 + m) as written.

Tests: ProductOfPowersDerivativeTest.ASumWithTheCommonPowersInEachTerm, the two rows above, differentiated back with the symbols pinned.

Measured first on all of Rubi's 1.1.3, 3,846 problems, at the corpus's 5-second budget, against master b1af529d (the cache's base; the change is cherry-picked there as 5ba12687):

	master	this
solved	3711	3713
unevaluated	114	111
unverifiable on the reals	5	5
past the budget	16	17

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	1319
families 2 to 8, sampled (2410)	2315	2320

The harness counts no answer wrong in either.

The 12 problems the two builds disagreed on, pocket and sample together, run again one build at a time: master answers 7, declines 4 and runs past the budget on 1; this answers 10, declines 1 and runs past the budget on 1.

The suite on the commit measured, 5ba12687 (the change cherry-picked onto b1af529d), passes, 14,987 tests, and so does the allocation gate: every gated benchmark allocates what the baseline says. The head here, 7ccac06a, merges master 15aae3d3; the calculus and corpus tests pass on it, 4,360 tests, and the library builds for netstandard2.0.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura