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x over a cubic binomial beside the root of another is integrated at the third elementary ratio - #1862
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Rafael-SOWNet merged 2 commits intoOct 12, 2026
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…he third elementary ratio Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
…e breaking changes Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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October 12, 2026 22:17
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Part of #718.
x/((a + b x^3) sqrt(c + d x^3))is elementary at three ratios, and the integrator knew two of them (#1790). This adds the third,a^2 d^2 - 20 a b c d - 8 b^2 c^2 = 0, Rubi's rules 57 and 58 in 1.1.3.4. Herer = (a d - 10 b c)/(6 b c)issqrt(3)or-sqrt(3). The antiderivative has two arctangents and two inverse hyperbolic tangents ins = sqrt(c r), and they swap places whenc r < 0. So the answer is split by that sign, and each branch is real.Measured on master f009b5d and on this branch (net10.0,
Integrate("x")):x/((b*x^3+2*a*(5+3*sqrt(3)))*sqrt(a+b*x^3))and its three sign variantsx/((b*x^3+2*a*(5-3*sqrt(3)))*sqrt(a+b*x^3))and its three sign variantsx/((10+x^3+6*sqrt(3))*sqrt(1+x^3))x/((4-x^3)*sqrt(1-x^3))x/((8-d*x^3)*sqrt(1+d*x^3))The corpus was run with both arms on master f009b5d. Every family is unchanged except for these 12 rows, which go from unevaluated to solved, with 0 wrong and 0 errors. 1.1.3.4:857 and 4.1.7:286 also moved, but they sit at the time limit and the changed code returns immediately for them. The gate passed, and the suite passed 15,282 tests. The unit test gains five rows, at both signs of
cand both signs ofr.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura