A perfect square written with symbols is read as one

[Rafael-SOWNet]

(A + B x)(d + e x)/(a^2 + 2 a b x + b^2 x^2)^(5/2) was left as written, though the radicand is (a + b x)^2 and the power is |a + b x|^5. Three conditions said no, and they are three different kinds of mistake:

1. The discriminant was not read symbolically. 4 a^2 b^2 - 4 a^2 b^2 is a zero InnerSimplified does not collect, so the guard read the square as an ordinary quadratic. (Third instance of this shape tonight, after An exponential times a trigonometric at the same frequency no longer divides by zero, and a phase is expanded #1472 and A symbolic constant factor no longer stops the rounds of parts #1473.)
2. The leading coefficient had to be a number. b^2 is not one, but it is positive for a real parameter — IsPositiveForARealParameter is the library's own answer to that, and its condition b^2 > 0 travels with the answer as it does elsewhere. Requiring positivity is right; requiring a literal is not.
3. My own complexity bound. The rule reads a radicand only up to a size, so that it costs nothing at every level of the descent — and a^2 + 2abx + b^2x^2 is over it. Lifted at the top, where the rule may work harder; below it, the square root only, for the reason A power of x beside a function of a symbolic power of x is integrated by substituting the power, and a square root of a perfect square is the modulus at every depth #1462 measured (a sign written for a substitution's variable is a factor the rest of the search carries: (1 + 2u + u^2)^(5/2) under u = e^(2x) cost twenty seconds).

And one defect the corpus found that no probe of mine had: with the sign written into the integrand, a rule below differentiates it, and sqrt(a^2 + 2abx + b^2x^2) sqrt(c + ex + dx^2) threw CannotEvalException: derivative(sgn(...)). The first measured run of this change reported 1 error where there had been none. The sign now goes in front of the integral, which is what every rule here that writes one does, and that row answers in half a second.

Measured (final build):

• Family 1 (familycap.sh 1 6): 156 → 158/228, 0 wrong, 0 error — 1.2.1.3:2008 and 1.2.1.6:99; one row moved Unsolved → Timeout at the 5 s cap ((d + ex)^(9/2)/(a^2 + 2abx + b^2x^2)^(3/2), which is a new answer path rather than a loss).
• Family 6: 377/417, 0 wrong, unchanged. 1774-problem suite: 1707, 0 wrong, 0 timeout, unchanged.
• Unit suite: 12,665 tests, 0 failed. bench.sh: PASSED on all 19 gated benchmarks.
• Four forms verified by differentiating back with every symbol pinned, worst relative deviation 0 at five points each.

Part of #718.

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