Determine all possible values on $\int_{\gamma} \frac{z^3}{(z+3)(z-5)^2} dz$ where $\gamma$ is a simple closed curve traversed once ccw and $-3$ and $5$ are not on $\gamma$.
So my intuition is to do partial fraction decomposition but that wouldn't be ~nice~. Obviously, the points $-3$ and $5$ are important and since it's traversed once ccw I'm assuming you can use the Cauchy-Gourzat theorem or the Cauchy integral formula but I don't know how to get there.
Since the residues of you function at $-3$ and at $5$ are $-\frac{27}{64}$ and $\frac{475}{64}$ respectively, the possible values are $0$, $2\pi i\times\left(-\frac{27}{64}\right)$, $2\pi i\times\frac{475}{64}$, and $2\pi i\times\left(-\frac{27}{64}+\frac{475}{64}\right)$,