A quadratic's half-odd power beside a power of another quadratic is integrated

[Rafael-SOWNet]

Part of #718.

Most of Rubi's 1.2.1.5 past the first powers was declined. That is a half-odd power of one quadratic B beside a whole power of another, A. There were two seams, one on each side of the bar:

• The root above the bar. SolveAPolynomialOverAPowerOfAQuadraticBesideTheRootOfAnother read only P/(A^k sqrt(B)). B^(n/2) above the bar, for an odd n, is now read as B^((n + 1)/2) over sqrt(B), which is a polynomial over the root. That covers sqrt(B)/A^2 and B^(3/2)/A^3.
• A power of the radicand below the bar. SolveARationalFunctionBesideTheRootOfAQuadratic splits the rational part over the written factors. TrySplitOverWrittenFactors declines a repeated quadratic with numbers in it, because over the rationals no rule reads a cubic over its square. Beside the root, though, each piece is a shape the rules read: a polynomial over the radicand's power is a polynomial times a half-odd power of the radicand, and one over the other quadratic's power goes to the reduction above. The root rule now says so through a new optional parameter, aRepeatedQuadraticIsRead. Every other caller is unchanged.

Sizes, measured on master 915f742, on 2.5.0 and on this branch:

Integrand	2.5.0	master 915f742	branch
sqrt(3-x+2*x^2)/(2+3*x+5*x^2)^2	unevaluated	unevaluated	1,344 chars
sqrt(3-x+2*x^2)/(2+3*x+5*x^2)^3	unevaluated	unevaluated	1,455 chars
(3-x+2*x^2)^(3/2)/(2+3*x+5*x^2)^3	unevaluated	unevaluated	1,451 chars
1/((3-x+2*x^2)^(3/2)*(2+3*x+5*x^2)^2)	unevaluated	unevaluated	1,430 chars
1/((3-x+2*x^2)^(3/2)*(2+3*x+5*x^2)^3)	unevaluated	unevaluated	1,552 chars
1/((3-x+2*x^2)^(5/2)*(2+3*x+5*x^2))	unevaluated	unevaluated	1,380 chars
1/((3-x+2*x^2)^(5/2)*(2+3*x+5*x^2)^2)	unevaluated	unevaluated	1,499 chars
1/((3-x+2*x^2)^(5/2)*(2+3*x+5*x^2)^3)	unevaluated	unevaluated	1,629 chars
(2+x)/((2+4*x-3*x^2)*(1+3*x-2*x^2)^(5/2)) (1.2.1.6)	unevaluated	unevaluated	804 chars
(2+x)/((2+4*x-3*x^2)*(1+3*x+2*x^2)^(5/2)) (1.2.1.6)	unevaluated	unevaluated	2,819 chars
(2+5*x+x^2)*sqrt(3+2*x+5*x^2)/(1+4*x-7*x^2)^3 (1.2.1.9)	unevaluated	unevaluated	5,242 chars
(a+b*x+c*x^2)^(1/2)/(d+e*x+f*x^2)^2	unevaluated	unevaluated	50,758 chars

The symbolic one is large, but it matches what master already gives with the root below the bar: 1/((d+e*x+f*x^2)^2*sqrt(a+b*x+c*x^2)) is 38,378 characters there.

Each was checked exactly, derivative against integrand at 50, 300 and 2,000 digits. Every one agrees at every precision, once #1868's fix (#1869, merged in) stops a rational from keeping the precision it was first written out at.

Rubi corpus, compared with master a29f7ec:

• Pocket of all of 1.2.1.4, 1.2.1.5 and 1.2.1.6: 847 → 858, with 0 wrong.
• 1.2.1.5 alone: 106 → 115 of 118.
• 13 problems moved. All 13 solve on this branch alone and 1 does on master alone. That one is 7.4.1:369, which takes 20–23 s on either arm against a 25 s timeout, so it is noise at the limit; it is the only difference in family 7's sample.
• Family 1's sample: 1347 → 1352.
• 0 wrong anywhere.

The allocation gate passed. The suite passed: 15,308 passed, 0 failed. Master 915f742 is merged in, and the build passes for every target framework.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura