Arms no real coefficients reach are left out of a symbolic quadratic's antiderivative

[Rafael-SOWNet]

Part of #718.

An integrand over a quadratic with symbols in it is answered by arms on the quadratic's leading coefficient and on the sign of its discriminant 4 a c - b^2, and two of those arms could not be reached:

integrand	2.5.0	master b1af529d	this
1/(1 + q x^2)	4 arms, 269 characters	4 arms, 269 characters	3 arms, 226 characters
1/(3 + q x^2)^2	4, 446	4, 446	3, 368
1/(x (a + q x))	4, 303, no value at a point where it has one	4, 303	3, 212
1/(x (a + q x))^2	4, 467, the same	4, 467	3, 315
1/(x (a + b x^n))	4, 417, the same	4, 417	3, 296

Each answer is differentiated back and compared with the integrand at six points.

What changes. IntegrateRationalQuadratic and IntegrateOverPowerOfQuadratic leave out:

• the arm for a zero discriminant where there is no linear term and the constant one is a number other than 0: the discriminant is 4 a c there, zero only where a is, which the arm before it takes -- and the arm left out divided by a;
• the arm for a positive discriminant where there is no constant term and the middle coefficient is real as written, symbols and real numbers by arithmetic and whole powers: the discriminant is -b^2 there. Real as written only: under sqrt(-(a + b x)^2) the middle coefficient is sqrt(-b^2) a/b, and the discriminant a^2 is positive; RootOfAPerfectSquareIntegralTest holds a row of that.

An answer is up to a third shorter, and so is every answer built on one: 1/(x (a + b x^n)) comes to 1/(u (a + b u)).

Tests: QuadraticDenominatorArmsTest, four rows with three arms each, differentiated back with the parameter positive and negative.

Measured first on Rubi's 1.1.2.2, 1.2.1.1 and 1.3.1, 1,300 problems, at the corpus's 5-second budget, against master 287c69a7, the branch's base:

	master	this
solved	1249	1249
wrong	0	0
past the budget	7	7

Measured then on the Rubi corpus against master 287c69a7:

	master	this
family 0, independent suites (1814)	1772	1772
family 1, 40 a file (1381)	1308	1305
families 2 to 8, sampled (2410)	2301	2293

The harness counts no answer wrong in either. The 11 problems the two builds disagreed on are the 11 the sample lost here, each past the budget; run again one build at a time, both builds answer the same seven of them and run past the budget on the same four.

The suite passes on the commit measured, 99ae5eee, 14,826 tests, and the allocation gate passed. The head here, a2a3750b, merges master b1af529d; the calculus and corpus tests pass on it, 4,280 tests, and the library builds for netstandard2.0.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura