# $\displaystyle\int_0^{\infty} \dfrac{\mathrm{d}x}{(x^4+ax^3+bx^2+cx+d)^m}$

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.

$$\int_0^{\infty} \dfrac{\mathrm{d}x}{(x^4+ax^3+bx^2+cx+d)^m}$$

• The bottom is a quartic to a power; since quartics are solvable exactly, you can turn this into a partial fractions problem. Not that this makes it that much less problematic; you'll have a quartic to factor, a huge matrix to solve, and then a bunch of individual integral terms to do up though fortunately those are powers of linears on the bottom. – Dan Uznanski Nov 4 '14 at 19:22
• You should specify conditions that guarantee the integral is convergent. For instance, if the polynomial is positive on $[0,\infty)$. Or if it does have zeros in $[0,\infty)$, that $m$ is an integer at least $2$ (which would make the integral break up into some convergent improper integrals). – alex.jordan Nov 4 '14 at 19:34
• @alex.jordan those are part of the solution. – UserX Nov 4 '14 at 19:40

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.