I am trying to compute$\int \frac{z^5 + z^3 + 19}{z^3 - 3z^2 +3z - 1} dz$ where $ \gamma$ is the circle of radius 7 centered at the origin. I have that $\gamma(t)= 7e^{it}$ so that $\gamma'(t)= 7ie^{it}$ for $t$ between 0 and $2 \Pi$. I have also factored the denominator to get $(z-1)^3$.
I have been trying to use $\int f(z) dz = \int f(\gamma(t))(\gamma '(t)) dt,$ but I keep getting strange answers that are taking a while to compute, so I'm not sure where I am going wrong. Should I be using another method?
By Cauchy's integral formula, if $f(z)=z^5+z^3+19$, then that integral is equal to$$2\pi i\frac{f^{(2)}(1)}{2!}=26\pi i.$$