It's a generalization for $$F(a,m)=\int_0^{\infty}\dfrac{\mathrm{d}x}{(x^4+2ax^2+1)^{m+1}}$$ that's being evaluated at Irresistible integrals, George Boros and Victor H. Moll 2004. I wonder if there exists a closed-form of it and the way to evaluate it, given that the first one is hard enough already.
$$\int_0^{\infty} \dfrac{\mathrm{d}x}{(x^4+ax^3+bx^2+cx+d)^m}$$
Hint: Compute $\displaystyle\int_0^\infty\dfrac{dx}{x^4+ax^3+bx^2+cx+d}$ and then differentiate m times with respect to d.