I am working on the integral of the reciprocal of polynomial such that
$\int{\frac{x^{q-1}dx}{x^{p}-ax^{q}+1}}$
where $p$ and $q$ are coprime integers.
I tried to solve it by partial fraction decomposition as below
$\int{\frac{x^{q-1}dx}{x^{p}-ax^{q}+1}}\\ =\int{\prod_{j=1}^{p}\frac{x^{q-1}}{(x-k_j)}dx}\\ =\int{\sum_{j=1}^{p}\frac{A_j}{(x-k_j)}dx} $
where $k_{1,...,p}$ are solutions of the polynomial $x^{p}-ax^{q}+1=0$.
Is there any rules that solves the coefficients $A_j$? or Is the existence of the coefficients $A_j$ guaranteed?