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Split a symbolic power of a product beside a symbolic power of one of its factors - #1887

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a-symbolic-power-of-a-product-splits-beside-a-power-of-a-factor
Oct 14, 2026
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Rafael-SOWNet merged 3 commits into
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a-symbolic-power-of-a-product-splits-beside-a-power-of-a-factor

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Part of #718

SolveBySplittingARootOfAProductBesideARootOfAFactor (#1876) wrote (P Q)^r as K P^r Q^r only for a rational r. K = (P Q)^r/(P^r Q^r) is exp(2 pi i n r) for a whole n that is constant between the zeros of P and Q, so the split holds for any exponent free of x. It now gathers symbolic exponents as well. In Rubi's 1.2.1.2:2389, (d + e x)^(-3 - 2p) (a d e + (c d^2 + a e^2) x + c d e x^2)^p, the radicand is (d + e x)(a e + c d x). Split, the powers of d + e x gather into (d + e x)^(-3 - p) (a e + c d x)^p, which master already answers.

Measured

intbench, this branch against master acd4702, 5 s budget. The pocket was 1.1.1.2–1.1.1.4, 1.2.1.2–1.2.1.4, 1.2.2.3 and 1.3.2 in full (10,159 rows), plus the usual family sample:

master this branch
pocket 10,019 solved 10,030 solved
families 0–8 sample identical
wrong / error 0 / 0 0 / 0

11 rows moved, all from unsolved to solved and all held when re-run alone; none went the other way:

  • 1.2.1.2:2389–2393
  • 1.2.1.3:2455
  • 1.2.1.4:979–982
  • 1.3.2:1355, (1 - x^4)^n/(1 + x + x^2 + x^3)^n

Every new answer was checked exactly with exactcheck (exact pins, 50 and 2,000 digits, four points, three pin sets): 240 of 240 diff 0. The gate and the suite pass (15,385).

The pins keep every exponent's denominator even. With p = 3/10 the check fails. The evaluator reads (-3.35)^(-18/5) as a real root but (-3.35)^(-3/10) (-3.35)^(-33/10) on the principal branch, so gathering powers of one base is not exact for it. Master's own normalisation does the same: (d + k x)^(-3 - 2p) (d + k x)^p (a k + c d x)^p fails there identically. I've added the measurement to #204.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura

Rafael-SOWNet and others added 3 commits October 14, 2026 14:47
(P Q)^r is K P^r Q^r with K constant between the zeros of P and Q for any
exponent free of x, not only a rational one: K is exp(2 pi i n r) for a
whole n.

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
…-a-product-splits-beside-a-power-of-a-factor
Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
@Rafael-SOWNet Rafael-SOWNet added this to the 2.6.0 milestone Oct 14, 2026
@Rafael-SOWNet
Rafael-SOWNet merged commit 364eda3 into master Oct 14, 2026
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