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SolveByWritingAPowerOfAQuotientApart(AngouriMath.Entity,AngouriMath.Entity.Variable)

Method (no overloads)

Summary

A fractional power of a quotient whose denominator is positive for every real
x is written as the power of the numerator over the power of the
denominator, a power of an even power of x among them as a power of x for x > 0, and the integrand so written is asked of the chain; where a
power of x was taken so, the answer is extended to x < 0 by parity.

Remarks

(P/Q)^r = P^r/Q^r whenever Q is a positive real, since then
arg(P/Q) = arg(P); a polynomial in x^2 with positive coefficients is one
at every real x but zero, where the quotient was undefined anyway. Timofeev's
((2 + x^2)/x^2)^(7/9)/(2 + x^2)^(3/2) is nothing any rule reads as written, and
x^(-14/9) (2 + x^2)^(7/9 - 3/2) once the quotient is written apart, a binomial
differential under u = x^(1/9). The simplifier is right not to write it so
for an x it knows nothing about; the integrator knows its variable is real.
(x^2)^(-7/9) is |x|^(-14/9), and is written as x^(-14/9): what comes
out is an antiderivative for x > 0, made one everywhere by parity, which is
exact, or not at all. Asked at the top and one level below it, where the remainder
by parts leaves is asked, and no deeper, since it lands on the open chain.
https://github.com/asc-community/AngouriMath/issues/718

AngouriMath

Table of contents

AngouriMath

SolveByWritingAPowerOfAQuotientApart​(AngouriMath.​Entity,​AngouriMath.​Entity.​Variable)

SolveByWritingAPowerOfAQuotientApart(AngouriMath.Entity,AngouriMath.Entity.Variable)