You use partial fractions to integrate rational functions of x (in other words, when you have a polynomial divided by another polynomial). And the best way to think about it is that it's a process that works - for ANY rational function. And I'm sure you're familiar with the process - first divide so you get a proper fraction, then factor the denominator and break apart into partial fractions, then evaluate the parameters, then integrate.
Each step is there to make it so the following steps work. So to answer your question, you use Bx+C when one of the factors of the denominator is an irreducible quadratic - an that doesn't factor into . And you should be able to integrate any function of the form . It might be messy, but you can always do it.