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A root of a square in a fractional power of x is the modulus as well - #1695
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Rafael-SOWNet merged 2 commits intoOct 2, 2026
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A sum of powers of x that are whole multiples of one fractional k is read as a polynomial in w = x^k, which is exact for every x, and the root of a square in it is answered for a positive x, where w is real, as the modulus of a linear in w. Rubi's 1.2.3.2 (a^2 + b^2/x^(2/5) + 2 a b/x^(1/5))^(5/2) is a square in x^(-1/5), and was declined: 0.5 s. A root of an even order plus a positive number, sqrt(x) + 1, is not given a sign. Part of #718. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
…ional-power-is-a-modulus # Conflicts: # Sources/AngouriMath/Functions/Continuous/Integration/IndefiniteIntegralSolver.cs
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…wer of x (#1698) * Any power of a square written out is the power of its root, in any power of x A square written out, A + B w + C w^2 with B^2 = 4 A C, was read as the modulus of its root only under half an odd power and in a whole power of x. A symbolic power, (a^2 + 2 a b x + b^2 x^2)^p, a power such as 3/4, and the square in x^n or x^(1/3) were declined. The power of the square is now the power of its root L = w + B/(2 C) times F = (A + B w + C w^2)^p / L^(2p), which is constant wherever L is not zero and comes out of the integral: F above the bar, 1/F below it, and nothing where the square is not a factor of the integrand. After the rule for half an odd power, which keeps its answers, and at the top only. On Rubi's 1.2.3.2, 179 fair problems whose trinomial is a square, 156 are answered where master answers 103, none wrongly. Rubi's 1.2.1.2, 1.2.2.2 and 1.2.3.2. Part of #718. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura * The square below the bar is tested where the root of a square does not answer it A half-odd power of a square in a fractional power of x is the root of a square's, by its modulus, since #1695. The two rows below the bar take a symbolic power and 3/4 instead, which only this rule answers. 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>
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Part of #718.
A root of a square written in a fractional power of
xis the modulus of a linear in that power. A sum of powers ofxthat are whole multiples of one fractionalk, such asa^2 + 2 a b x^(1/3) + b^2 x^(2/3), is read as a polynomial inw = x^k, which is exact for everyx. The root of a square in it is answered as the modulus of a linear inw, as the root of a square quadratic is since #1672.wis real for a positivex, and the answer says so,provided x > 0:A root of an even order plus a positive number, such as
sqrt(x) + 1, is not given a sign.These answers differentiate back only where their condition holds, which #1685 makes possible. Rubi's 1.2.3.2
(a^2 + b^2/x^(2/5) + 2 a b/x^(1/5))^(5/2), #659, is a square inx^(-1/5). Master declines it and this branch answers it in half a second.Tests:
RootOfAPerfectSquareIntegralTest.ASquareInAFractionalPowerOfTheVariable, four rows. Each is differentiated back under its condition witha bof either sign, so thatw + a/bchanges sign among the points.Measured first on Rubi's 1.2.3.2, the 179 fair problems whose trinomial is a square, beside #1685, which this branch was stacked on before it merged. Both arms differentiate an answer with its condition:
Then on the corpus against master
c3f52691, the branch's base, with both builds side by side:No answer is wrong on either build. Three problems moved between the two runs, so I re-ran each alone on both builds. This branch answers two of them, 1.2.3.2's #659 and 3.2.1's #76, and master one, #76, in about 21 s on each. So the family-3 problem that moved the other way is near the budget on both. Neither build answers the third, 4.4.10's #45.
The suite passes, 14,382 tests on a fresh build, and so does the allocation gate. Master
96ac3bc7is merged in since. Its one conflict was #1691's method added beside this one, and both are kept; the tests of both pass on the merge.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura