complex analysis - Determine residues of $frac{e^{-sqrt{z(z+r))}}}{1+alphasqrt{z(z+r)} + (1-alpha sqrt{z(z+r)})e^{(-sqrt{z(z+r)})}}$

I have tried to determine residues of the below function via Mathematica and Matlab, but they lead me nowhere. For small enough $\alpha$, I figured out what are the poles, but nothing about residues. My method was somehow straight-forward, using Taylor series.
Does anybody have an idea to calculate its residues?

$\frac{e^{-\sqrt{z(z+r))}}}{1+\alpha\sqrt{z(z+r)} + (1-\alpha \sqrt{z(z+r)})e^{-2\sqrt{z(z+r))}}}$

Thanks in advance.