I've learnt from a few YouTube videos that in the case of integrals of functions of the form $x^m(ax^n+b)^\frac{p}{q}$, there are a three standard substitutions you can make based on those exponents. My confusion is with the case where $\frac{p}{q}\in\mathbb{Z}$.
From my understanding, if that exponent is an integer, one method is either to simply use binomial expansion and then solve the integral or to make the substitution $x=t^z$, where $z$ is the smallest common multiple between the denominators of $m$ and $n$.
That being said, I have two questions I did not find an answer for. What is the standard method if $\frac{p}{q}\in\mathbb{Z}$ is negative? We have not learned how to using binomial expansion with negative powers, so I would not be supposed to use that if it is even possible. Lastly, if the method of using the smallest common multiple as I have mentioned is indeed correct, what does one do when $m$ and $n$ are integers, such that the recommended substitution would become $x=t$?