What is the integral or maybe a suggested technique to find $$\int_0^x \left(1+\left(\frac{t}{b}\right)^a\right)^{-q} d t,$$ $a,b,q\in \mathbb{R}:a\cdot q>1,b>0.$ ?
Note that there will not, in general, be an elementary antiderivative. For example, if $a=3$, $b=1$, $q=1/2$, you are asking for $$\int_0^x{dt\over\sqrt{1+t^3}}$$ which is an example of an elliptic integral, a so-called "special function", and not expressible in closed form in terms of rational functions, exponentials, logarithms, trig and inverse trig functions. For other values of the parameters you probably get functions that don't even have names, much less closed form expressions in terms of familiar functions.