x over a cubic binomial beside the root of another is integrated where that is elementary

[Rafael-SOWNet]

Part of #718.

x/((a + b x^3) sqrt(c + d x^3)) is an elliptic integral except at two ratios of its coefficients, 4 b c = a d and 8 b c + a d = 0, where it is arctangents and inverse hyperbolic tangents of sqrt(c + d x^3) and of (1 + q x)/sqrt(c + d x^3) for q = (d/c)^(1/3). Welz's x/((4 - x^3) sqrt(1 - x^3)) and the rest of Rubi's 1.1.3.4 at those ratios were declined:

integrand	2.5.0	master 221d3c5f	this
x/((4 - x^3) sqrt(1 - x^3))	declined	declined	390 characters
x/((4 - d x^3) sqrt(-1 + d x^3))	declined	declined	481 characters
x/((4 c + d x^3) sqrt(c + d x^3))	declined	declined	1,165 characters
x/((8 + x^3) sqrt(-1 + x^3))	declined	256 characters	230 characters
x/((8 - d x^3) sqrt(1 + d x^3))	declined	declined	337 characters
x/((8 c - d x^3) sqrt(c + d x^3))	declined	declined	897 characters

Each is differentiated back at six points with the symbols pinned.

What changes. SolveXOverACubicBinomialBesideTheRootOfAnother, asked after SolveAPseudoEllipticQuotientOverTheRootOfACubicBinomial, reads x/((a + b x^3) sqrt(c + d x^3)) and a constant multiple of it, and writes Rubi's closed forms with y = sqrt(c + d x^3) and q the real cube root:

• at 4 b c = a d, four terms, q/(3 2^(2/3) b r) times artanh(y/r)/3 + atan(y/(sqrt(3) r))/sqrt(3) - atan(sqrt(3) r (1 + 2^(1/3) q x)/y)/sqrt(3) - artanh(r (1 - 2^(1/3) q x)/y) for r = sqrt(c) and a positive c, and for a negative one, with r = sqrt(-c), the same with each arctangent and inverse hyperbolic tangent exchanged and the first two signs turned;
• at 8 b c + a d = 0, where the integrand is -(d/b) x/((8 c - d x^3) y), three, (artanh(r (1 + q x)^2/(3 y))/18 - artanh(y/(3 r))/18 - atan(sqrt(3) r (1 + q x)/y)/(6 sqrt(3)))/(r^3 q^2), and for a negative c the same exchanged with the first and last signs turned.

Rubi reaches the second by splitting the integrand into three integrals, one a substitution's and two of the kind #1781 answers. Asked as three, two come back piecewise in the symbols and their sum holds every combination of the arms: 5,607 characters for x/((8 - d x^3) sqrt(1 + d x^3)), where the closed form is 349. Every condition is decided as a value, the form is chosen by the sign of c -- both, each where it holds, for a symbol -- and the answer is differentiated back at sampled points before it is returned.

Tests: XOverACubicBinomialBesideTheRootOfAnotherIntegralTest, eight rows at the two ratios, with c of either sign, differentiated back on both sides of 0 wherever the integrand is real.

Measured first on all of Rubi's 1.1.3.4 and Welz's problems, at the corpus's 5-second budget, against master b1af529d, the branch's base:

	master	this
solved	740	746
unevaluated	61	55
wrong	0	0
past the budget	5	5

Measured then on the Rubi corpus against master b1af529d:

	master	this
family 0, independent suites (1814)	1775	1778
family 1, 40 a file (1381)	1316	1318
families 2 to 8, sampled (2410)	2315	2313

The harness counts no answer wrong in either.

The 16 problems the two builds disagreed on, pocket and sample together, run again one build at a time: this answers six that master does not -- Welz's 150, 151 and 153, and 1.1.3.4:319, 324 and 364 -- the ratios above. Four that master answered alone ran past the budget here, 1.1.2.4:343 and 548, 3.2.1:152 and 4.1.2.3:76; run once more with a minute each, they take 20 to 26 seconds on either build, and the load decides them. The other six are answered by both, declined by both or past the budget on both.

The suite on the commit measured, df3fc1d3, passes but for AnIntegralWithNoAnswerStillFinishesQuickly, whose 30-second bound took 33 seconds on a machine running two corpus chains; alone it takes two to three seconds, with this rule or without it. The allocation gate passed: every gated benchmark allocates what the baseline says. The head here, cc7d132d, merges master 221d3c5f; the calculus and corpus tests pass on it, 4,334 tests, and the library builds for netstandard2.0.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura