I encountered the following identity in a paper on number theory,

$$\int_{-\infty}^{\infty}\frac{dW}{(W+i)^{\frac{3}{2}}(W^2+1)^s}=\frac{e^\frac{-3\pi i}{4}\sqrt{2}\pi \Gamma(2s+\frac{1}{2})}{2^{2s}\Gamma(s+\frac{3}{2})\Gamma(s)},$$

with $Re(s) > 0$ and $i=\sqrt{-1}$.

Since the author did not give the proof for this, maybe it is "well-known", but I failed to gave a proof. Note that the right hand side looks like the Beta function with some multiplier, and we can write $W^2+1=(W+i)(W-i)$ on the left hand side, maybe we need to do some variable substitution.

Does any one know how to prove this identity or have some references? Thank you for any kind of input.