Split a power of a product beside a whole power of a factor, and ask it early

[Rafael-SOWNet]

Part of #718

SolveBySplittingARootOfAProductBesideARootOfAFactor (#1876, #1887) split (P Q)^r into K P^r Q^r only when the factor beside it was a root or a symbolic power too. It now also splits in two more cases:

• A whole power of a factor, beside a root of the product. Rubi's 1.2.1.2:2218, 1/((d + e x)^4 (a d e + (c d^2 + a e^2) x + c d e x^2)^(3/2)), gathers into (d + e x)^(-11/2) (a e + c d x)^(-3/2). Only for a factor to a whole power of two or more below the bar:
  • Beside a bare factor, Welz's (1 + x)/((x - 2) sqrt(1 + x^3)) became three logarithms instead of one, ALinearDividingARadicandOfSymbols hung the suite, and 1.1.1.2:1257 went from 4 ms to 36 s.
  • Above the bar, (a + b x)^3 sqrt(c x^2 - a^2 c/b^2) went from 3,540 characters to 33,316.
• A whole power of the product, or the product standing bare, beside a root of a factor that is a sum. For example 1.2.1.2:2280, 1/(sqrt(d + e x) (a d e + (c d^2 + a e^2) x + c d e x^2)). Here K is 1. Beside a power of x, x^(5/2)/(b x^2 + c x^4)^3 went from 1.7 s to 7.9 s.

Two whole powers are a rational function and are left to the partial fractions.

The rule now runs right after SolveByWritingEachLinearOnce, ahead of the routes that read the product whole. Placed late, 2218 took 27 s to reach it, and 2276 was answered by another route first in 29,285 characters. Now they take 0.5 s and 1,611 characters.

Measured

intbench, this branch against master 364eda3, 5 s budget. The pocket was 1.1.1.2–1.1.1.4, 1.2.1.2–1.2.1.4, 1.2.1.6, 1.2.1.9, 1.2.2.2–1.2.2.4 and 1.3.2 in full (13,826 rows), plus the usual family sample:

	master	this branch
pocket	13,674 solved, 52 timeouts	13,683 solved, 47 timeouts
time on the rows both solve	1,899 s	1,847 s
rows more than twice as slow		none
families 0–8 sample		identical
wrong / error	0 / 0	0 / 0

Nine rows moved, all to solved, and all held when re-run alone (master solves none of them alone):

• 1.2.1.2: 1081, 1091, 2218, 2228, 2275, 2276 and 2280
• 1.2.1.3: 2447 and 2531

All nine were checked exactly with exactcheck (exact pins, 50 and 2,000 digits, four points, three pin sets): 216 of 216 diff 0. The gate and the suite pass (15,391).

A wider version also answered 1.2.1.3:147 and 156, (A + B x)/(x^4 (b x + c x^2)^(3/2)). It paid for that with the 3,540-to-33,316 growth and the 7.9 s above, so I left those two rows out.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura