Hot Integral - 25

Calculus Level 5

$\displaystyle \int \limits_0^\infty \int \limits_0^\infty \int \limits_0^\infty \int \limits_0^\infty \frac{1}{wxyz \left(w+x+y+z+\dfrac1w + \dfrac1x + \dfrac1y + \dfrac1z \right)^2} \ dw \ dx \ dy \ dz$

If the integral above is equal to $\dfrac{A}{B} \zeta(C)$ , where $A$ and $B$ are coprime positive integers and $C$ is an integer, find $A+B+C$ .

Notation: $\zeta(\cdot)$ denotes the Riemann zeta function.