An inequality involving two probability densities

I cannot prove the following inequality, which I state below:

Let $p, q$ be two positive real numbers such that $p+q=1$. Let $f$ and $g$ be two probability density functions. Then, show that:

$$\int_{\mathbb{R}} \frac{p^2 f^2 + q^2 g^2}{pf + qg} \geq p^2+q^2~.$$

I tried to use Cauchy-Schwarz and even Titu's lemma, but got nowhere. Any help will be greatly appreciated. Thanks!