A power of the variable below the bar beside a root of a quadratic

[Rafael-SOWNet]

1/(x^2 sqrt(4 + 3x + 2x^2)) was left as written while 1/(x sqrt(...)) came out — the single factor was read and the repeated one was not. The repeated one is exactly what a round of parts against arccos(a + b x)/x^4 leaves, so a family-5 row came with it.

Under x = 1/t the integrand is -sgn(t) t^(n - 1)/sqrt(q0 t^2 + q1 t + q2): a polynomial over the root of the quadratic read the other way round, which the rules for those already answer (0.3 s).

Three details that decided the implementation, each measured:

• The radicand is assembled, not substituted into. Q(1/t) is R(t)/t^2, so the root is R^(p/2) |t|^(-p); writing the quotient inside the root leaves a nesting nothing downstream reduces — the same trap as in A root of a quadratic in the tangent is rotated until the quadratic has no linear term #1475.
• The modulus is a sign in front. sqrt(R/t^2) is sqrt(R)/|t|, so an odd power leaves sgn(t), which at t = 1/x is sgn(x) — what the answers of this family carry anyway.
• The reciprocal's own t != 0 had to be stripped. InnerSimplified attaches it as a Providedf, and a condition is not an integrand to the rules below: with it attached, 1/(x^2 sqrt(...)) declined while 1/(x^3 sqrt(...)) came out, which is what put me on to it.

SolveByReciprocalSubstitution is this substitution for a different shape — a palindromic quartic under the root, where the reciprocal maps the quartic to itself — and reads nothing here.

Measured (final build):

• Family 5 (familycap.sh 5 15): 211 → 212/257, 0 wrong. Family 1 (familycap.sh 1 6): 155 → 156/228, 0 wrong. 1774-problem suite: 1707, 0 wrong, 0 timeout, unchanged.
• Unit suite: 12,662 tests, 0 failed. bench.sh: PASSED on all 19 gated benchmarks.
• Six forms verified by differentiating back at five points — numeric and symbolic coefficients, n from 2 to 4 — worst relative deviation 0 on every one.

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura