Solve with Cauchy Integral theorem: $\oint_C \frac{1}{z(2z+1)^2} \, \text{d}z$ but solve with partial fractions

Question

I want to solve this integral $\oint_C \frac{1}{z(2z+1)^2}\, \text{d}z$ with parcial fractions using Cauchy Integral theorem. There's no other way to solve without being partial fractions? Trying to put a $2z$ in evidence and then move on to the numerator being $\frac{1}{2z}$ and so using Cauchy theorem?

C is a unitary circle