This question seemed very trivial at the first glance, but evidently it is a non-trivial problem, at least, for me.
Let $p, q$ be polynomials in $x$, and $\deg q > \deg p$, with $q (x)$ being irreducible, and there is no constant $A$ such that $q ' (x) = A p (x)$. Then how do we find $$I (x) = \int \frac {p (x)} {q (x)} dx?$$ I cannot even calculate the integral for individual pairs of $(p (x), q (x))$, let alone for the general case.
I tried decomposing the given fraction to such parts whose anti-derivatives are known, but it became more complicated than the original problem. Then I tried integrating by parts, which resulted in terribly tedious and far more complicated integrals. I tried using expansions, i.e., for some function $y (x) \in (-1, 1)$ I tried differentiating, integrating or doing some other operations several times the expression $$\frac {1} {1 - y} = \sum y^k$$ to arrive at $p (x) / q (x)$, but all to no avail.
I think either I'm stupid or I'm unaware of a theory regarding this, or there's indeed no method. Thanks in advance.