Read the root's quadratic at a symbolic pole without the slope's condition

[Rafael-SOWNet]

Part of #718

sqrt(a + c x^2)/(x (d + e x)) and its kin in Rubi's 1.2.1.4 were declined, though sqrt(a + b x + c x^2)/(x (d + e x)) was answered. The difference was in SolveALinearBesideTheRootOfAQuadratic, which works at the linear's root p = -d/e. With no b x term, (a p^2 + c).InnerSimplified comes back as 3 (-d/k)^2 + 2 provided not k = 0. TryStandardIntegrals reads no quadratic in t through that Providedf, so 1/((d + k x) sqrt(2 + 3 x^2)) was declined where 1/((d + x) sqrt(2 + 3 x^2)) was not. The condition only says the linear's slope is not zero, so the value is now read with Bare, here and in SolveAPolynomialOverAPowerOfALinearBesideTheRoot.

The condition had also been stopping the rule on linears with the imaginary unit in them. With it gone, 1 + i a x (what e^(3 i atan(a x)) is written as) reached the rule as a pole at i/a. The rule then wrote sgn(x - i/a) and nothing checked it afterwards, and 5.3.6:69, 1/(e^(3 i atan(a x)) x^3), lost its answer. The rule now declines a linear holding the imaginary unit, which is what it did by accident before. The other rule checks its own derivative, and master's route for 5.3.6:69 goes through it, so it keeps complex poles.

Measured

intbench, this branch against master 1763843, 5 s budget. The pocket was 1.2.1.2–1.2.1.4, 1.2.1.6, 1.2.1.9, 1.2.2.4, 1.3.2, 5.3.6, 7.3.6 and 4.3.2.1 in full (9,239 rows), plus the usual family sample:

	master	this branch
pocket	9,029 solved	9,044 solved
families 0–8 sample		identical
wrong / error	0 / 0	0 / 0

All 15 moves held when re-run alone (master solves 1 of them), and none went from solved to anything else:

• 1.2.1.4: 404–408, 416–418, 424–426, 439 and 441
• 1.3.2:1233, sqrt(a + c/x^2)/(d + e x)
• 4.3.2.1:690

I checked the 14 algebraic rows exactly with exactcheck (exact pins, 50 and 2,000 digits, four points, three pin sets with negative coefficients): 336 of 336 diff 0. The gate and the suite pass (15,389).

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura