A quadratic's coefficients are named for the odd half power's solve

[Rafael-SOWNet]

Part of #718.

sqrt(a + b x + g^2 x^2/f^2) is answered on master after three seconds, in 188,000 characters, where sqrt(a + b x + g^2 x^2) is answered at once in a few hundred: the reduction of a polynomial times an odd half power of a quadratic solves one linear system in the quadratic's coefficients, and a quotient or a product of symbols among them was carried through the elimination as written. 2.5.0 declined these:

integrand	2.5.0	master cb8e133d	this
sqrt(a + b x + g^2 x^2/f^2)	declined	188,008 characters, in 3.5 s	712, in 0.5 s
x/sqrt(a + b x + g^2 x^2/f^2)	declined	39,401, in 1.0 s	657, in 0.6 s
(a + b x + c x^2/f)^(3/2)	declined	477,711, in 10.4 s	933, in 0.4 s
x^2 sqrt(a f + b x + c x^2/f)	declined	152,336, in 3.5 s	1,140, in 0.5 s

The times are of the integration alone, each in a fresh process; differentiating master's answers back to check them took up to a minute and a half more.

What changes. Each coefficient that is neither a number nor a symbol is named for the solve, and written back into the solution before anything else reads it; the table's integral and the final check stay on the radicand as written.

Tests: CompoundCoefficientsOfAQuadraticIntegralTest, five rows differentiated back with the symbols pinned, and held to an answer of under 5,000 characters, which is what the length of the answer measures where a test cannot hold a time; on master the five are answered in 40,000 to 480,000.

Measured first on the 333 problems of family 1 with a root of a quadratic whose coefficients have a quotient in them, at the corpus's 5-second budget, against master 8f3757cd, the branch's base:

	master	this
solved	220	223
wrong	0	0
past the budget	5	2

Measured then on the Rubi corpus against master 8f3757cd:

	master	this
family 0, independent suites (1814)	1766	1766
family 1, 40 a file (1381)	1296	1297
families 2 to 8, sampled (2410)	2215	2214

No answer is wrong in the pocket or the sample on either build. Of the 11 problems the two builds disagreed on, pocket and sample together, run again one build at a time, master answers 2 and this 7. Three are the 1.3.2 problems that put a compound coefficient under the root, (d + e x + f sqrt(a + b x + e^2 x^2/f^2))^k for k from one to three, past the budget on master and answered here in 0.02 to 9 seconds. The other four this answers, and the two master does, take about twenty seconds on either build, at the edge of the harness's patience; two of them, x/csch(x)^(3/2) + x sqrt(csch(x))/3 and (a + b arctanh(c x^2))^2/x^5, are the one each families 6 and 7 lose in the sample, and both are answered here run alone. The last four are past the budget on both builds.

The allocation gate passes on the commit measured, cb1cd28a. Its suite run there was stopped by the machine's memory guard at 8 GB after 9,812 tests had passed and none failed: the chain runs the suite with no heap limit, and the guard watches the test host. On the merge with master cb8e133d, d722c7e4, the whole suite under a 4 GB heap limit passes, 14,780 tests with 13 skipped, and every row of the first table is as it says.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura