Pretty Nice Integral

$\displaystyle \int_{0}^{\infty} \dfrac{x^{5}\mathrm{d}x}{e^{5x} -1} = \dfrac{1}{a^{b}}\zeta(c)\Gamma(c)\\$

With $a,b,c$ are positive integers with prime number $a$, find $\dfrac{5}{6}abc^2$

$\text{ Details and Assumptions }$
1.)$\zeta(x)$ is the Riemann Zeta Function
2.)$\Gamma(x)$ is the Gamma Function