A rational function of x^n beside a power of x and the root of a binomial is integrated

[Rafael-SOWNet]

Part of #718.

x/((a + b x^3)^(2/3) (c + d x^3)) and x^4 (a + b x^3)^(1/3)/(c + d x^3), Rubi's 1.1.3.4, were declined, while the same without the power of x in front is answered by dividing by the root, under u = x/(a + b x^n)^(1/n). 2.5.0 declined them too:

integrand	master 959c2d2e	this
x/((a + b x^3)^(2/3) (c + d x^3))	declined	0.8 s
x^4 (a + b x^3)^(1/3)/(c + d x^3)	declined	6.7 s
x/((1 + x^3)^(2/3) (2 + x^3))	declined	0.1 s
x^7 (a + b x^3)^(1/3)/(a d - b d x^3)	declined	0.5 s

What changes. x^j times a rational function of x^n times (a + b x^n)^(k - (j + 1)/n) is u^j times the same in u, since x^j (a + b x^n)^(-j/n) is u^j; SolveByDividingByTheRoot read j = 0 only, and reads any j below n now, with x^j taken out of the polynomial above the bar first. Each answer holds on both sides of zero, an odd root of a negative being real here.

Tests: DividedByTheRootIntegralTest.BesideAPowerOfX, five rows, each differentiated back with the symbols pinned on both sides of zero; all five fail on master.

Measured first on all of Rubi's 1.1.3.3 and 1.1.3.4, 941 run, at the corpus's 5-second budget, against master 8f3757cd, the branch's base:

	master	this
solved	851	860
wrong	0	0

Measured then on the Rubi corpus against master 8f3757cd:

	master	this
family 0, independent suites (1814)	1766	1766
family 1, 40 a file (1381)	1296	1302
families 2 to 8, sampled (2410)	2215	2219

No answer is wrong in the sample on either build. Of the 21 problems the two builds disagreed on, run again one build at a time, 15 are answered here and not on master: 11 of 1.1.3.4's of this shape, three of 1.1.2.2's -- (c x)^(5/2)/(a - b x^2)^(3/4) and two powers of (c x)^(7/3) beside a cube root of a + b x^2 -- and one of 6.6.1's that ran out of time on master. Two went the other way in that run, x^4/((a + b x^2)^2 (c + d x^2)^3) and (a + b x^3)^(2/3)/(x^12 (a d - b d x^3)); both take 26 to 34 s on either build without a budget, measured alternately, at the edge of the corpus's. The other four are answered on both.

The suite passes on the commit measured, d4165a5d, 14,596 tests, and so does the allocation gate. Master 959c2d2e is merged in since, without conflicts; the calculus tests pass on the merge, 4,009 of them.

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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura