[Rafael-SOWNet]

Integrates a polynomial beside a function of one shifted linear a + b x that parts cannot reach, by writing it in u = a + b x. Part of #2075.

The shapes:

• a power that is not whole of something holding a logarithm of the linear, x (a + b ln(c (d + e x)))^p;
• an exponential of a function of it that is not linear, x^2 f^(c (a + b x)^3).

In u the polynomial is one in u, and each of its terms beside the function is a gamma function of #2083's rules. This is the substitution SolveAPolynomialBesideErrorFunctionsOfAShiftedLinear already makes for error functions. In u the offset is zero, so the rule declines there and cannot recurse on its own output.

	Rubi	in u
x (a + b ln(c (d + e x)))^p	3.3	(u - d) (a + b ln(c u))^p / e^2
x (a + b ln(c (d + e sqrt(x))))^p	3.4	after the root substitution, a cubic in u
x^2 f^(c (a + b x)^3)	2.3	((u - a)/b)^2 f^(c u^3) / b

The function is found by its shape first, before any factor is read as a polynomial, which expands it. The first version read the other way round, and asked of every product the algebraic files are, it was 5 to 10% of the corpus's time with no allocation to show for it. Read this way, time per stream is 0.97 to 1.01 of master's.

Measured on Rubi's suite, read with the harness that maps GAMMA, against master 3dbae246, the whole corpus each side:

	master	this
solved	48,875 of 49,052	48,907
wrong	0	0
solved rows lost		0
rows to a timeout		0

The gains are 18 in 3.4, 9 in 2.3 and 5 in 3.3. No answer's allocation grew.

By value:

	this	quadrature
integral(x*(1/3 + 2*ln(3/2*(1 + 2*x)))^(5/2), x, 1, 2)	47.0187310513846589616701…	47.01873105138465896167011
integral((1/3 + 2*ln(3/2*(1 + 2*sqrt(x))))^(5/2), x, 1, 2)	24.8155926461197117937625…	24.81559264611971179376254
integral(x^2*2^((1 + x/2)^3), x, 0, 1)	2.2029698172036954849810…	2.202969817203695484981035

The suite passes (16,944) and so do the MCP tests (329). The new tests differentiate four shapes back, one of them with a constant in front, which the first version dropped.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura