Can integrals of this type be computed?

Question

I know that integrals like this can be computed using complex analysis:

$$\oint_C \frac{e^z dz}{z^{2}- 2z - 3} $$

but can the following be integrated using complex analysis?

$$\oint_C \frac{e^z dz}{z^{\alpha}- 2z - 3} $$

EDITED: To be clear, $\alpha \in \mathbb{R}$.