I would like to evaluate the following complex integral, \begin{equation} \oint_C \frac{f(z)}{z} \, \log z \, \text{d}z\,, \end{equation} where $f(z) = a + b/z - \bar{b}z $ with $a,b \in \mathbb{C}$. The contour $C$ is a circle surrounding a line joining the origin $0$ and one of the roots of $f$, denoted $z_+ =\frac{a-\sqrt{a^2+4|b|^2}}{2\bar{b}}$. So the circle encloses a first branch cut, and there are two other branch points (the second root of $f$, $z_-$, and the point at infinity) outside the circle.
One problem (for me at least) seems to be that the point $z=0$ is a pole of the integrand and also lies at the end of the inner cut, so one cannot directly integrate along the cut.
My knowledge of contour integral is rather limited once one cannot use residue theorem (nor residue at infinity), and I would make great use of some help on the procedure to follow to deal with this kind of integrals.