The third case of a binomial differential is written back for a negative x too

[Rafael-SOWNet]

Part of #718.

Chebyshev's third case of a binomial differential, x^m (a + b x^n)^(p/q) with (m + 1)/n + p/q whole, is integrated under x = 1/y, and wrote its root back as (b + a/x^n)^(1/q). That is the root for a positive x, and for a negative one only under an odd root, which is real here. Under an even root the answer was wrong for every negative x:

integrand	master dbe5b382	this
1/(1 + x^4)^(5/4)	1/(1 + 1/x^4)^(1/4), an even function	1/((1 + x^4)^(1/4)/x)
x^2/(1 + x^4)^(3/4)	logarithms and an arctangent of (1 + 1/x^4)^(1/4)	the same of (1 + x^4)^(1/4)/x
x^6 (3 + 4 x^4)^(1/4)	the same in (4 + 3/x^4)^(1/4)	the same in (3 + 4 x^4)^(1/4)/x

Each integrand is real on the whole line. Checked exactly in sympy at x = -0.71, -0.43, 0.37, 1.31: on master the derivative of the answer minus the integrand is -2f at the two negative points and 0 at the positive ones; here it is 0 at all four.

What changes. The root is written back as (a + b x^n)^(1/q)/x^(n/q), whose q-th power is b + a/x^n at every x, so the answer is right on both sides of zero under either root. The case came in with #1288, after 2.5.0, which declined these three.

Tests: BinomialDifferentialTest.TheThirdCaseOnBothSidesOfZero, six rows real on both sides of zero, each differentiated back at three negative and three positive points. On master the three under a fourth root fail.

The corpus does not see this. It compares at positive points, and at negative ones only where it cannot compare at positive ones. work/intbench has a switch for it now, IB_BOTHSIDES=1, which also compares at the negatives of the positive points. On master it counts three of Timofeev's problems wrong this way, (3 + 4x^4)^(1/4)/x^2, x^2 (3 + 4x^4)^(5/4) and x^6 (3 + 4x^4)^(1/4), each answered with the integrand's negative for a derivative at every negative x; here all three are right on both sides, checked with the library's own settings at five negative and three positive points.

Measured first on every fourth problem of Rubi's 1.1.2.2 and 1.1.3.2, 794 run, at the corpus's 5-second budget, against master dbe5b382, the branch's base:

	master	this
solved	745	745
wrong	5	5

The five wrong on both are powers of x whose exponent is -1 as a value and not as written, such as x^(-1 - 3n) (a + b x^n)^3, which this does not touch.

Measured then on the Rubi corpus against master dbe5b382:

	master	this
family 0, independent suites (1814)	1766	1767
family 1, 40 a file (1381)	1292	1292
families 2 to 8, sampled (2410)	2209	2209

No answer is wrong on either build. The one problem the two builds disagree on, run again one build at a time, is Timofeev's cos(x) (-cos(x)^2 - 5 sin(x)^2)^(3/2), complex-valued on the whole line: on master its answer differs from the integrand at a sample point and is left unverifiable, and here it checks out.

The suite passes on the commit measured, 764400de, 14,594 tests, and so does the allocation gate. Master 3a3d0163 is merged in since, without conflicts, and the calculus tests pass on the merge, 3,965 of them.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura