Symbolic factors of a denominator that share a factor are written over it

[Rafael-SOWNet]

Part of #718.

The partial fractions over factors with symbols in them read the written factors as coprime and squarefree. The refactoring after them reads rational coefficients only. So a quadratic that shares a root with a linear beside it, or that is a square, was read as an irreducible quadratic:

integrand	master	written over what the factors share	this
1/((d + e x)(a d e + (c d^2 + a e^2) x + c d e x^2)^2)	declined after 3.8 s	1/((d + e x)^3 (a e + c d x)^2), 74 ms	0.4 s
(d + e x)^6/(a d e + (c d^2 + a e^2) x + c d e x^2)^4	past 30 s	(d + e x)^2/(a e + c d x)^4, 11 ms	2.6 s
1/((d + e x)^2 (c d^2 + 2 c d e x + c e^2 x^2)^3)	declined	1/(c^3 (d + e x)^8), 5 ms	0.1 s
x^2/((a + b x)(a^2 + 2 a b x + b^2 x^2))	declined	x^2/(a + b x)^3	0.04 s

These are Rubi's 1.2.1.2, (d + e x)^m (a d e + (c d^2 + a e^2) x + c d e x^2)^p, whose quadratic is (d + e x)(a e + c d x), and its squares.

What changes. Before the symbolic split, the written factors are taken apart over their greatest common divisors until they are coprime and squarefree, using the multivariate gcd the library already has:

• each pair of factors over what they share;
• each factor over what it shares with its derivative in x;
• a factor over what it shares with the numerator, which is then cancelled.

A part a divisor leaves that is free of x goes in front whole, with its sign: a divisor is found up to a unit, and -(b x^2 - 1)^2 taken apart over its derivative leaves -1. A common factor that is a power of x alone is not split over. Taking a power of x out of a factor is the content's, which runs before this, and split here it wrote the rest of the factor monic with a symbol below its bar: Rubi's 1.1.4.3 x^12 (A + B x^2)/(b x^2 + c x^4)^3 came to an answer of four million characters, where the content's way gives seven thousand.

The divisors are taken over the rationals in every variable, so whatever of the symbols alone comes with one goes in front. Each factor is written monic in x, as the integrator writes one. Written with whole coefficients, the next normalisation took the leading coefficient out again, and the answer carried e^(-8) e^8. A coefficient with a symbol below a bar is put over one bar first, so a/c + (c d^2 + a e^2)/(c d e) x + x^2, which is how the integrator writes the quadratic, is read too.

SolveByCancellingWithFunctionsAsIndeterminates has a squarefree factorisation of its own, but only for a denominator with a function of x in it. It reads the denominator as one product, so factors of the same multiplicity come back multiplied together. Here the written factors stay apart.

Tests: SymbolicFactorsPartialFractionsTest.FactorsThatShareOneAreWrittenOverIt, the four rows above, each differentiated back with the symbols pinned; all four fail on master. And TheNegativeOfASquareKeepsItsSign, three rows, Rubi's 1.1.2.2 x^4/(a + b x^2)^(3/2) among them.

Measured on the Rubi corpus against master 94ba5ff6. The master column was measured beside an earlier commit of this branch; the branch column is this commit, run afterwards over the same samples:

	master	this
family 1, 40 a file (1381)	1117	1121
family 0, independent suites (1814)	1757	1758
families 2 to 8, sampled (2586)	2363	2362

Both builds give the same two wrong answers in family 1, 1.2.1.1's roots of negative squares, which #1672 fixed on master after this branch's base, and the same one in family 7, which #1682 fixed.

27 problems moved between the two runs, and on a machine this loaded that includes problems near the budget either way, so I re-ran each alone on both builds. This branch answers 19 of them and master 7, neither wrongly, and none that master answers is lost.

The suite passes, 14,319 tests on a fresh build, and so does the allocation gate.

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