A whole power of a sum of two square roots below the bar is rationalised

[Rafael-SOWNet]

Part of #718.

x/(sqrt(a + b x) + sqrt(c + b x))^3 was declined, while the first power of such a sum below the bar is multiplied above and below by its conjugate, the product of the pair being the difference of the radicands. 2.5.0 declined them too:

integrand	2.5.0	master 287c69a7	this
x/(sqrt(a + b x) + sqrt(c + b x))^3	declined	declined	powers of the two roots, in 4 s
1/(sqrt(a + b x) + sqrt(a + c x))^3	declined	declined	the same with logarithms and arctangents, in 7 s
x^2/(sqrt(a + b x) + sqrt(a + c x))^3	declined	declined after 10 s	the same, in 3 s
x^3/(sqrt(a + b x) + sqrt(a + c x))^2	declined	answered in 13 s	the same, in 0.2 s

The times include differentiating the answer back with the symbols pinned to check it.

What changes. A whole power of the pair below the bar is multiplied by the same power of the conjugate, expanded, over the same power of the difference of the radicands. An even power over a constant difference is left as it was: its conjugate brings the root of the product of the radicands with it, a root of a quadratic, and the squares over sqrt(a + b x) + sqrt(c + b x) went past the budget so where they are answered in a second as written.

Tests: APowerOfASumOfRootsIntegralTest, four rows differentiated back with the symbols pinned; three are declined on master.

Measured first on the 17 problems with a power of a sum of two square roots below the bar, all of them run, at the corpus's 5-second budget, against master 8f3757cd, the branch's base:

	master	this
solved	9	16
wrong	0	0
past the budget	2	1

Measured then on the Rubi corpus against master 8f3757cd:

	master	this
family 0, independent suites (1814)	1766	1767
family 1, 40 a file (1381)	1296	1296
families 2 to 8, sampled (2410)	2215	2217

The harness counts no answer wrong in the pocket or the sample on either build. Of the 17 problems the two builds disagreed on, pocket and sample together, run again one build at a time, master answers 4 and this 11: eight that master declines, seven of 1.3.2's powers of a sum of roots and Welz's 1/(sqrt(x^2 - 1) (sqrt(x) + sqrt(x^2 - 1))^2) from the independent suites, answered here in two and a half to eight seconds. The other nine are at the edge of the harness's patience or beyond the budget on both: two both answer at nineteen to twenty-four seconds, two that master answers at twenty-one seconds ran past the patience here, one past it there is answered here at twenty-four, and four are declined or past the budget on both.

The suite passes on the commit measured, b51e3148, 14,595 tests with 13 skipped, and the allocation gate with it. On the merge with master 287c69a7, dc7bcd54, the 4,141 calculus and corpus tests that run pass, with 2 skipped. Every row of the first table is as it says on the merge.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura