Suppose $F(x) = P(x)/Q(x)$ is an integrable rational function on $\mathbb R$, that is, $\deg Q \geq \deg P + 2$, and $Q$ has no real roots.

Does there exist an expression for the definite integral $I_F = \int_{-\infty}^\infty F(x)dx$ in terms of only coefficients of $P$ and $Q$?

So far my idea was to apply the the residue theorem and the standard semicircle contour. This expresses $I_F$ as a symmetric function of half of all roots of $Q$, namely, those in the upper half-plane. Could we somehow upgrade it to a symmetric function of all of the roots, and thus the function of coefficients?