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A quadratic with rational roots takes Euler's third substitution at one of them - #1872
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Rafael-SOWNet merged 4 commits intoOct 13, 2026
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…ne of them Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
…tional-roots-takes-eulers-third-substitution # Conflicts: # Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs
Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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Part of #718.
1/(x + sqrt(3 - 2x - x^2))was declined. The leading coefficient is negative and the constant term is not a square, so Euler's second substitution wrotesqrt(3)into every coefficient and the rational integrator did not finish. The quadratic is(1 - x)(3 + x). Shifted to its larger rational root,x = y + r, its constant term is zero and the third substitution answers with rational coefficients.The shift is taken only after the first substitution is ruled out, that is, where
ais not positive. Taken before it, it caughtsqrt(12x^2 + 17x + 6)/((2 + 3x)^2 (30 + 31x - 12x^2)^2)(1.2.1.6:218), which the first substitution answers in about 1 s, and timed it out.Measured against master 234e45e and v2.5.0:
1/(x+sqrt(3-2x-x^2))1/(x+sqrt(3-2x-x^2))^21/(x+sqrt(3-2x-x^2))^3x/(x+sqrt(3-2x-x^2))1/(2x+sqrt(3-2x-x^2))sqrt(6+17x+12x^2)/((2+3x)^2 (30+31x-12x^2)^2)The new answers were checked against exact values at 50 digits: their derivatives were compared with the integrand at sample points between and outside the radicand's roots. The new test,
ARationalRootOfTheQuadratic, pins four of them.The Rubi corpus, all families, against master 683ab15 (the branch's base before the merge): 3 moved, all from unsolved to solved (1.3.2:1000–1002), with 0 wrong answers on either side. The allocation gate passes. The suite passes: 15,320 passed, 0 failed.
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