How can the correct form of the partial fractions decomposition be found for arbitrary rational functions? What is the reasoning or intuition that leads to the assumption that 
$$r(x) =\frac{x^2 + 2}{ (x+2)(x-1)^2}$$ 
can be expressed as 
$$r(x) = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+2}$$
(For the sake of context, this problem arose when trying to solve an integral by the method of partial fractions.)
Moreover, how can the correct form of the partial fractions decomposition be found for other rational functions?

It's not an assumption. What you're trying to do is rewrite the fraction on the left as a sum of fractions on the right. Since the antiderivatives of the fractions on the right are easier to compute, the internal is easier to take. 
For every factor, one gets a fraction. If the factor is linear, the numerator is a constant. If the factor is an irreducible quadratic, then the numerator is a linear term. If the factor has multiplicity $m$, then we get m fractions, one for each power.
