I am stuck at this slight variation of the usual integration of Laurent series. I have the following Laurent series of function $f(z)$ $$f(z)=\frac{8z+9}{(2z+1)(z+3)} = \sum_{n=0}^\infty {\frac{(-3)^n}{(z-1)(n+1)}}+\sum_{n=0}^\infty {\frac{3(-1)^n(z-1)^n}{4^{n+1}}}$$
Now, I need to find the contour integral
$$\int_C \frac{8z^2+9z}{(z-1)^3(2z+1)(z+3)}dz=\int_C \frac{f(z).z}{(z-1)^3}dz$$ where C is the circle |z-1|=2 oriented in the counterclockwise direction.
I would have been able to solve it if it's simply $f(z)$. Now that it is multiplied by $z$, I am clueless how to approach the question.