A fractional power of a quotient whose denominator is positive for every real x is written as the two powers it is, and the integrand so written is asked

[Rafael-SOWNet]

Timofeev's ((2 + x^2)/x^2)^(7/9)/(2 + x^2)^(3/2) is nothing any rule reads
as written, and x^(-14/9) (2 + x^2)^(7/9 - 3/2) once the quotient is
written apart -- a binomial differential under u = x^(1/9), which the
linear radical substitution and the binomial rule answer between them in
half a second. (P/Q)^r = P^r/Q^r whenever Q is a positive real, since
then arg(P/Q) = arg(P), and a polynomial in x^2 with positive
coefficients is one at every real x but zero, where the quotient was
undefined anyway. The simplifier is right not to write it so for an x it
knows nothing about; the integrator knows its variable is real.

(x^2)^(-7/9) is |x|^(-14/9), and is written as x^(-14/9): what comes
out is an antiderivative for x > 0, made one everywhere by parity --
exact, since an integrand with only even powers of x in it is even and
its antiderivative odd -- or not at all. Both sides are checked. Asked at
the top only, as every rule that lands on the open chain is, and beside
the rule that combines radicals, which does the like for the roots it
combines.

Rubi sample (463 problems, 5 s budget), against the #1315 master this branch is cut from:

master at #1315     406/463, 0 wrong, 0 timeouts, ~100 s
with this           407/463, 0 wrong, 0 timeouts, 98 s

One more, nothing lost: Timofeev 642, in 0.5 s. No other problem moved by more than noise. Independent of #1316 (open), which touches the same two files in other places; the raw diff between the two runs shows only each other's problem.

The five test suites are green.

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura