x^2 over a three-quarter power of a quadratic binomial beside another is integrated where that is elementary

[Rafael-SOWNet]

Part of #718.

x^2/((A + B x^2)^(3/4) (C + D x^2)) at B C - 2 A D = 0, Rubi's 1.1.2.4, is the difference of the two functions whose sum answers 1/((A + B x^2)^(1/4) (C + D x^2)) at that ratio, with another constant, and was declined; 2.5.0 declined it too:

integrand	master 959c2d2e	this
x^2/((a + b x^2)^(3/4) (2 a + b x^2)), a and b positive	declined	0.7 s
the same, b negative	declined	0.8 s
the same, a negative	declined	0.7 s
x^2/((-2 + 3 x^2) (-1 + 3 x^2)^(3/4))	declined	0.2 s
x^2/((2 - 3 x^2)^(3/4) (4 - 3 x^2))	declined	0.2 s

What changes. The rule that answers 1/((A + B x^2)^(1/4) (C + D x^2)) at B C = 2 A D reads x^2 above the bar and (A + B x^2)^(3/4) below it as well, and answers it by the signs of A and B: an arctangent less an inverse hyperbolic tangent, over A^(1/4) and the cube of the root of B or -B -- of -A where A is negative, where the two share one argument -- times B/D, as the quarter power's answer is. The constants are solved for from the two functions' derivatives, and each form is checked at points in its sign case.

Tests: EllipticLookingQuotientOfBinomialsTest.XSquaredOverTheThreeQuarterPower, six rows across the sign cases of A and B, each differentiated back at points in its case, the negative side among them; all six fail on master.

Measured first on the 446 problems of Rubi's 1.1.2.3 and 1.1.2.4 with a quarter power in them or in their answers, 196 run, at the corpus's 5-second budget, against master 8f3757cd, the branch's base:

	master	this
solved	142	160
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	1298
families 2 to 8, sampled (2410)	2215	2218

No answer is wrong in the sample on either build. Of the 25 problems the two builds disagreed on, run again one build at a time, 20 are answered here and not on master, 18 of them 1.1.2.4's of this shape, and four are answered on both. One went the other way in that run, 1/((a g + b g x)^3 (A + B ln(e (a + b x)/(c + d x)))), which has no quarter power in it and takes 35 to 43 s on either build without a budget, measured alternately, at the edge of the corpus's.

The suite passes on the commit measured, c1fe3336, 14,597 tests, and so does the allocation gate. Master 959c2d2e is merged in since, without conflicts; the calculus tests pass on the merge, 4,010 of them.

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https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura