An exponential of a polynomial beside the polynomial's derivative is integrated under u = P

[Rafael-SOWNet]

Part of #1501.

G^P k P' f(P), where P is a polynomial of degree two or more and P' is a factor of its own up to a constant, is now integrated under u = P, as k G^u f(u). Rubi's 2.3 has nine of these, e^(a + b x + c x^2) (b + 2 c x) (a + b x + c x^2)^(n/2). Beside the exponential, u^(n/2) is answered already, as F^(a + b x) x^(n/2) is.

The substitution search reads this u and does not reach it. For its own search it writes an exponential of a sum as a product of exponentials, so that e^(e^x) e^x becomes e^u du. But e^(a + b x + c x^2) then becomes e^a e^(b x) e^(c x^2), with no P left in it to replace. On master the search declines e^(a + b x + c x^2) (b + 2 c x) sqrt(a + b x + c x^2) after 24 s, where 2.5.0 declined it in under 2. The new rule answers it in 270 ms, from before the search.

Only a derivative written as a sum is taken, so x^2 e^(x^3) stays with the rules that answer it already.

integral	master	this PR
e^(a + b x + c x^2) (b + 2 c x) (a + b x + c x^2)^(n/2), n = 1, 3, 5, 7	unevaluated, 3 past 3 s	answered, with erfi
the same over (a + b x + c x^2)^(n/2), n = 1, 3, 5, 7, 9	unevaluated	answered, with erfi

Measured

Measured with the change on 12ee9809 (#1520), before the rebase onto #1521 and #1522. After the rebase, the library builds for every target, and the Gaussian, exponentials-of-quadratics and exponential-substitution test classes pass (175). The erf baseline is 22021b1c, one merge earlier, and the 29 problems #1520 gained are counted apart, by name.

• Rubi's problems whose answer uses erf, erfc or erfi: 811 across families 0 to 7, at a 3 s budget. This PR solves 631, where 22021b1c solves 593 and An exponential of a quadratic in the reciprocal of a linear is integrated onto the Gaussian #1520 adds 29, with 0 wrong and nothing lost. The 9 this rule gains are the 9 of its shape, each answered in 3 to 8 ms, and 3 of them were timeouts, so timeouts go from 14 to 11. The total time goes from 574 s to 544 s.
• The textbook suites (family 0) are unchanged: 1723/1778, 0 wrong, 0 timeouts.
• The unit suite passes on net10.0: 12952 passed, 14 skipped, none failed, of 12966.
• The allocation gate passes on all 19 gated benchmarks, with no allocation moved.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura