Evaluate a real integral, e.g. $\int_{0}^{\infty}\frac{x^2}{(x^2+1)(x^2+4)}\:\mathrm dx$ with complex analysis

My question is more theoretical, i.e, i cannot quiet understand the "method" itself.

For example $$\int_{0}^{\infty}\frac{x^2}{(x^2+1)(x^2+4)}\:dx$$ I know that the fact of the denominator having no real roots and that $deg((x^2+1)(x^2+4))-deg(x^2)\geq2$ are "important informations" but i dont how to apply them. Can someone help me? I know that we are supposed to apply the residue Theorem after, but i cannot understand the steps in order to apply it.

Thank you!

from Hot Weekly Questions - Mathematics Stack Exchange