A power of a multiple of a quadratic's derivative beside a power of the quadratic is a binomial

[Rafael-SOWNet]

Part of #718.

Rubi's 1.2.1.2 has 344 problems of the form (b d + 2 c d x)^m (a + b x + c x^2)^p, a power of a multiple of the quadratic's derivative beside a power of the quadratic. With a power that is not whole these were declined or ran out of time, since nothing saw that the linear is a multiple of the derivative.

Under t = b d + 2 c d x the quadratic is (t^2/d^2 - (b^2 - 4 a c))/(4 c). The substitution is affine, so this holds exactly, and the integrand is a binomial in t. The new rule, SolveByTheDerivativeOfAQuadraticAsTheVariable, does that:

• It writes the quadratic in t from its coefficients. Substituting instead would leave a linear term that is zero as a value and not as written.
• It names each constant of the binomial that is a compound of the coefficients by a symbol of its own, and writes it back into the answer. The binomial rules answer t^(-7) sqrt(c t^2 + k) in 63 ms, and with a - b^2/(4 c) written for k they ran past twenty seconds.
• It leaves a quadratic with no linear term alone, since that is a binomial already, and taken in t it would be the same question with t scaled.

integrand	master	this
(a + b x + c x^2)^(3/2)/(b d + 2 c d x)^3	declined	0.8 s
(a + b x + c x^2)^(1/2)/(b d + 2 c d x)^7	declined	0.8 s
(a + b x + c x^2)^(5/2)/(b d + 2 c d x)^5	declined	0.8 s
(b d + 2 c d x)^(5/2)/(a + b x + c x^2)^3	declined after 26 s	0.9 s

Master is e33fbab2, and the times include checking the answer at six points.

On the whole 1.2.1.2 pocket against master be009d56, the 234 of those 344 problems whose answers are in functions the library has, each run alone at the corpus's 5-second budget: 160 solved on master, 211 here, none wrong on either, and 13 timeouts where master has 17.

Tests: DerivativeOfAQuadraticAsTheVariableTest, the four rows above, differentiated back with the symbols pinned, on both sides of the derivative's root wherever the integrand is real.

Measured on the Rubi corpus against master be009d56, with both builds side by side:

	master	this
family 0, independent suites (1814)	1757	1758
family 1, 40 a file (1381)	1113	1117
families 2 to 8, sampled (2586)	2362	2362

Both builds give the same three wrong answers: two in family 1, 1.2.1.1's roots of negative squares, which #1672 fixed on master after this branch's base, and one in family 7, e^(2 acoth(a x)) sqrt(c - a c x)/x, which #1682 fixed.

Six problems moved between the two runs, so I re-ran each alone on both builds. This branch answers three that master declines or runs out of time on, and none that master answers is lost: from 1.2.1.2, sqrt(c d^2 + 2 c d e x + c e^2 x^2)/(d + e x)^4, sqrt(a + b x + c x^2)/(b d + 2 c d x) and 1/((b d + 2 c d x)^3 (a + b x + c x^2)^(3/2)).

The suite passes, 14,316 tests on a fresh build, and so does the allocation gate. Master e33fbab2 is merged in since, without conflicts; the four rows above were measured on that merge, and its tests pass.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura