[Rafael-SOWNet]

Part of #233.

• Whole powers of a root written as powers of its radicand. Under u = sqrt(x), Rubi's 1.1.3.2 1/(x^(7/2) sqrt(a + b x^5)) came back with x^5 spelled ((x^(1/2))^5)^2, under provided x >= 0. The same happened to three like it, to 1.3.2's sqrt(a x^23)/sqrt(1 + x^5) and sqrt(a x^13)/sqrt(1 + x^5), and to 1.1.1.2's 1/(sqrt(x) sqrt(2 x - 3)).
• Why the condition stayed. The integrand is real beyond -(a/b)^(1/5), as a product of two imaginary roots, and the answer holds there. But WhereTheDerivativeHoldsBeyondTheRadicand evaluates ((x^(1/2))^5)^2 at a negative x with a zero imaginary part, on the outer root's cut, and could not say so.
• The change. WithWholePowersOfARootAsPowersOfItsRadicand flattens a whole power of a root and folds (B^(1/q))^n, where q divides n, into B^(n/q), which holds on every branch. It is applied to the substitution's back-substituted answer and at the start of ConditionedOnTheRadicand. The condition is gone, and the answers are shorter.

Rubi row	master ef4150e	this branch
1.1.3.2:1462 1/(x^(7/2) sqrt(a + b x^5))	92 characters, for x >= 0 only	60 characters, both sides of 0
1.1.3.2:1463 1/(x^(17/2) sqrt(a + b x^5))	134, for x >= 0 only	97
1.1.3.2:1469 1/(x^(7/2) sqrt(1 + x^5))	82, for x >= 0 only	50
1.1.3.2:1470 1/(x^(17/2) sqrt(1 + x^5))	112, for x >= 0 only	75
1.3.2:511 sqrt(a x^23)/sqrt(1 + x^5)	563, for x >= 0 only	426
1.3.2:512 sqrt(a x^13)/sqrt(1 + x^5)	388, for x >= 0 only	293
1.1.1.2:1791 1/(sqrt(x) sqrt(2 x - 3))	183, for x >= 0 only	141

"For x >= 0 only" is the probe at -2.3 and -1.7: the answer has no value at 2 of 5 points where the integrand has one.

Chain against master 78cd31b:

• The pocket of 12,081 rows went from 12,060 to 12,081 solved.
• Every family's sample is unchanged.
• The 21 rows that moved, run alone, all went from partial to solved. Fourteen of them Integrate a power of x beside a binomial in an odd negative power of x on both sides of 0 #2038 had already fixed on master since.
• 17,596 rows were compared for allocation, and none grew past 1.3x and 300 MB.

The full unit suite ran locally with 16,552 passed and 1 failed. The failure was IntegralAnswerCacheTest.AnIntegralWithNoAnswerStillFinishesQuickly at a load of 17, and it passes alone in 2 s. The BREAKING-CHANGES lengths were measured on the branch. Rebased onto ef4150e.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura