A polynomial with symbols in it that is a binomial or an even quartic in a shifted variable is integrated in it

[Rafael-SOWNet]

Part of #718.

A polynomial below the bar with a symbol among its coefficients was declined whenever it is not already written in the form a rule reads, although some are one shift of the variable away from it: c^2 x^3 + 3 b c x^2 + 3 b^2 x + 3 a b is ((c x + b)^3 + 3 a b c - b^3)/c, and 1/(a + (b + c x)^3) is answered at once.

integrand	master 599dcb7f	this
1/(3 a b + 3 b^2 x + 3 b c x^2 + c^2 x^3)	declined	0.5 s
1/(3 a b + 3 b^2 x + 3 b c x^2 + c^2 x^3)^2	declined after 1.6 s	0.6 s
x/(a + 8 x - 8 x^2 + 4 x^3 - x^4)	declined	0.4 s
1/(a + 8 x - 8 x^2 + 4 x^3 - x^4)	declined	0.4 s

What changes. One step in the partial fractions, before the Hermite reduction. Where the denominator is a constant times a power of one polynomial of the third to the sixth degree with a symbol in it, written out, the polynomial is read in y = x + s with s = a_(n-1)/(n a_n), which takes its term of degree n - 1 away. Where that leaves a binomial in y, or a quartic even in y -- a + 8 x - 8 x^2 + 4 x^3 - x^4 is a + 3 - 2 y^2 - y^4 in y = x - 1 -- the integral is taken in y and written back. A polynomial over the rationals is not touched, since the steps above split it over its factors, and neither is one written in a linear already, a + (b + c x)^3.

Tests: ShiftedPolynomialIntegralTest, four rows, each differentiated back with the symbols pinned.

Measured first on Rubi's 1.3.1 Rational functions, all 490 problems with answers in functions the library has, at the corpus's 5-second budget, against master 9c2509d8, the branch's base:

	master	this
solved	437	451
wrong	0	0

14 more are answered and none that master answers is lost.

Measured then on the Rubi corpus against master 9c2509d8:

	master	this
family 0, independent suites (1814)	1766	1766
family 1, 40 a file (1381)	1287	1289
families 2 to 8, sampled (2410)	2197	2197

No answer is wrong on either build. The 14 problems the two builds disagreed on, run again one build at a time, are all answered here and none on master, all in 1.3.1.

The suite passes on the commit measured, b46c2bd5, 14,563 tests, and so does the allocation gate. Master 599dcb7f is merged in since, without conflicts, and the calculus tests pass on that merge, 3,939 of them; 2340e8a4, #1711, after it, with the tests the two touch passing.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura