Calculate a complex integral by its residues

Question

I am working on a problem that is related to probability and statistics. I need to find the Mellin transform of a function for which I know the Laplace transform. I know that the relation between Laplace and Mellin transform is $$ M_{r}(f_{y}(y))=\frac{\Gamma(r)}{2\pi i}\int_{c-i\infty}^{c+i\infty}L_{s}(f_{y}(y))(-s)^{-r}ds, $$ where $M_{r}(f_{y}(y))$ and $L_{s}(f_{y}(y))$ are the Mellin and Laplace transform of $f_{y}(y)$. The Laplace transform is $$ L_{s}(f_{y}(y))=\frac{1}{\sqrt{(1+s)(1+2s)}}\exp(\frac{-\alpha^{2}s}{1+s}), $$ where $\alpha$ is a constant. So, how can I compute the following integral? $$ M_{r}(f_{y}(y))=\frac{\Gamma(r)}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{1}{\sqrt{(1+s)(1+2s)}}\exp(\frac{-\alpha^{2}s}{1+s})(-s)^{-r}ds. $$ Can I find the result by computing the residues of the function? If YES, how can I find the residues at relevant poles?