Consider the integral $\int_{\mathbb{R}}f(x^{\frac{1}{m}})dx$ where $f$ is smooth and has compact support and $m$ is odd.
Can one change variables $y=x^{\frac{1}{m}}$ ?
My concern is about the negative values of $x$. If $x$ is a real negative number, then we know that $y$ does take a negative real value. But this is precisely one of the $m$ different values of the $m$th root of $x<0$. The other $m-1$ roots are all complex with a nonzero imaginary part. Note that if $m$ is even then $y^m=x$ has $m$ complex roots none of which is real.