I've found a simple formula for this integral, but it uses p1, q1, p2, q2 such that P(x)= a4 ( x^2 + p1 · x + q1) · ( x^2 + p2 · x + q2).
These values are always real, but I don't know how to find them.

Maybe there is a solution which doesn't use them?

Of course there is.
1). Use Ferrari method to find roots.
2). Use numerical method to find roots.
3). Use generic numerical methods without finding roots.
All are not-so-easy to implement.