How to verify whether the abnormal integral is convergent or divergent?

Question

How to verify the convergence and divergence of the abnormal integrals $\int_{0}^{1} \frac{\sqrt[m]{\ln ^{2}(1-x)}}{\sqrt[n]{x}} d x$:

Integrate[Power[Log[1 - x]^2, (m)^-1]/Power[x, (n)^-1], {x, 0, 1}, 
 Assumptions -> Element[m | n, PositiveIntegers]]

No results can be obtained by running the above code. I don’t expect to get the integral value of this integral related to m, n. I just want to judge whether it converges (and the relationship between convergence and m, n). What should I do?

Answer 1

The result of

Series[Power[Log[1 - x]^2, (m)^-1]/Power[x, (n)^-1], {x, 0, 2}]

$$ \left(x^2\right)^{1/m} x^{-1/n} \left(1+\frac{x}{m}+\frac{(5 m+6) x^2}{12 m^2}+O\left(x^3\right)\right)$$ implies the convergence condition -1/n + 2/m > -1 at the origin and the result of

Series[Power[Log[1 - x]^2, (m)^-1]/Power[x, (n)^-1], {x, 1, 2}, Assumptions -> x < 1]

$$(-\log (1-x))^{2/m}\left(1-\frac{x-1}{n}+\frac{(n+1) (x-1)^2}{2 n^2}+O\left((x-1)^3\right)\right. $$ means that the integral converges at $x=1$ for any positive $m$.