Mellin transform q series it is posible to calculate 
$$\int_0^{\infty } \frac{x^{z-1}}{q^x-1} \, dx$$
as Mellin transform Mathematica could not do it

Substitute  $t=x\log q$ to get $$\int_0^{\infty} \frac{1}{\ln^{z}q}\frac{t^{z-1}}{e^t-1}dt=\Gamma(z)\zeta(z)\frac{1}{\ln^{z}q}$$
