A power of a monomial in a symbolic power of x is integrated as the power of x it is

[Rafael-SOWNet]

Part of #718.

1/(a + b (c x^n)^(2/n)) and the rest of Rubi's 1.1.3.2 with a power of a monomial in a symbolic power of x, (c x^n)^(k/n), were declined, where the same integrand with c^(k/n) x^k in its place is answered:

integrand	2.5.0	master 2b86349c	this
1/(a + b (c x^n)^(2/n))	declined	declined	506 characters
1/(x^2 (a + b (c x^n)^(1/n)))	declined	declined	159 characters
1/(a + b (c x^n)^(3/n))	declined	declined	656 characters
1/(1 + 4 (x^(2n))^(1/n))	declined	declined	arctan(2 x)/2, 29 characters

Each answer is differentiated back and compared with the integrand at six points.

What changes. DistributeAFractionalPower writes (c x^n)^b as c^b x^(n b), and left the exponent n b as the inner simplification leaves it: (c x^n)^(2/n) came out as c^(2/n) x^(n (2/n)), which no rule reads as x^2. The exponent is simplified now where it is not a number and is small, so that x^(n (2/n)) is x^2.

Tests: DistributedPowerOfAMonomialIntegralTest, four rows with a symbolic n, one of them (x^(2n))^(1/n), differentiated back.

Measured first on all of Rubi's 1.1.3.2, 2,466 problems, at the corpus's 5-second budget, against master 287c69a7, the branch's base:

	master	this
solved	2404	2416
wrong	0	0
past the budget	0	0

Measured then on the Rubi corpus against master 287c69a7:

	master	this
family 0, independent suites (1814)	1772	1771
family 1, 40 a file (1381)	1308	1305
families 2 to 8, sampled (2410)	2301	2296

The harness counts no answer wrong in either. The 21 problems the two builds disagreed on, pocket and sample together, run again one build at a time: this answers 12 that master does not, 1.1.3.2:3708-3709, 3715-3716, 3732-3734, 3739-3740 and 3746-3748, and master none that this does not. The other nine take over 5 seconds on either build or run past the budget on both, and the load decides whether they are answered: 3.2.3:29, answered here by this and not by master, master answered in 19 seconds in another run.

The suite passes on the commit measured, 5956ceee, 14,826 tests, and the allocation gate passed. The head here, ea40df5a, merges master 7269344b; the calculus and corpus tests pass on it, 4,312 tests, and the library builds for netstandard2.0.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura