Complex integral with non integer power of the denominator

Question

I would like to solve this complex integral but I'm not able to use Residue Theorem because of the non integer power of the denominator. Do you have a trick for this problem ?

$$\int_{c_2 - i \infty}^{c_2 + i \infty} \frac{z e^z}{(\alpha-z^2)^\frac{3}{2}}\, dz$$

$\alpha$ and $c_2$ are positive and real numbers.

Any help would be appreciated,

Thanks