Evaluating an integral using the residue theorem

Question

I'm having trouble evaluating the complex integral \begin{equation} \int_\gamma \frac{1}{|z-\alpha|^2|z-\beta|^2z}dz, \end{equation} where $\gamma$ refers to the unit circle, the integral being taken anticlockwise, and $\alpha$ and $\beta$ are constants with $|\alpha|<1$ and $|\beta|<1$. The integrand has singularities at $\alpha$, $\beta$, and 0. I've already computed the residual of the integrand $f$ at 0, and got $Res(f,0) = \frac{1}{|\alpha|^2|\beta|^2}$. I can't figure out the residuals at $\alpha$ and $\beta$. They don't seem to be simple poles, since the limit \begin{equation} \lim_{z->\alpha}\frac{z-\alpha}{|z-\alpha|^2|z-\beta|^2z} = \lim_{z->\alpha}\frac{1}{(\overline{z}-\overline{\alpha})|z-\beta|^2z} \end{equation} does not exist.