An even root of a quotient with an odd power below the bar keeps its sign below zero

[Rafael-SOWNet]

Part of #718.

sqrt((1 + x)/x^3) is real for x < -1 as well as for x > 0, and master integrates it as sqrt(1 + x) x^(-3/2), which it is for x > 0 only: below -1 the two differ by a sign, and the answer is wrong at every point there. 2.5.0 declined these:

integrand	2.5.0	master 9588be9e	this
sqrt((1 + x)/x^3)	declined	wrong at every x < -1	sgn(x) times the answer for x > 0, right on both sides
sqrt(x/(1 + x)^3)	declined	wrong at every x < -1	the same with sgn(1 + x)
((1 + x)/x^3)^(3/2)	declined	wrong at every x < -1	the same as the first
sqrt((2 + x)/(1 + x)^3)	declined	wrong at every x < -2	the same with sgn(1 + x)

What changes. The check that decides whether an even root of a quotient may be written apart summed the multiplicities of the factors that can be negative on both sides of the bar alike, so 1 + 3 = 4 passed for (1 + x)/x^3, where the phase of the split is above minus below, 1 - 3 = -2. A split now needs both sums to be multiples of twice the root's index, which refuses exactly the splits that were wrong and keeps every other. Where the root does not come apart, a linear's odd power below it comes out with its sign in front of the answer, |1 + x| being sgn(1 + x) (1 + x).

Tests: RootOfAQuotientOnBothSidesIntegralTest, five rows differentiated back at points on both sides, at least two of them below zero and three above; all five are wrong on master.

Measured first on the 254 roots of quotients in the corpus, 191 run, at the corpus's 5-second budget, against master 8f3757cd, the branch's base, and again with the harness checking every answer at the negatives of its points as well:

	master	this
solved	170	170
wrong	0	0
solved, both sides checked	170	170
wrong, both sides checked	0	0

The corpus writes these roots with symbols in them, which go through a rule that already takes the sign out, so it holds none of the shapes that were wrong; the tests and the first table do.

Measured then on the Rubi corpus against master 8f3757cd:

	master	this
family 0, independent suites (1814)	1766	1766
family 1, 40 a file (1381)	1296	1296
families 2 to 8, sampled (2410)	2215	2218

No answer is wrong in the sample on either build. The six problems the two builds disagreed on have no root of a quotient in them, and run again one build at a time each took about twenty seconds on either, at the edge of the corpus's patience: three came out on master's side of it and none on this one's, which is the machine and not the change -- two of them, measured alone with a minute's budget, take 22 and 24 s on both builds.

The suite passes on the commit measured, b833c76d, 14,596 tests with 13 skipped, and the allocation gate with it. On the merge with master 9588be9e, 03bb47e0, the 4,076 calculus and corpus tests pass, and every row of the first table is as it says.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura