a question about Cauchy integral formula

I'm new in the complex analysis and I'm stuck with this integral :

$I=\displaystyle \int_{|z|=4} \frac{\mathrm{d}z}{(z^2+9)(z+9)} $

the exercise is about Cauchy integral, I don't want the whole solution, just give me a hint (Please don't post fully worked solutions)

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what I have done : using partial fraction we get :

$I=\displaystyle \frac{1}{90} \int_{|z|=4} \frac{1}{9+z} + \frac{(9-z)}{9+z^2} \mathrm{d}z$

I'm trying to do this : $\displaystyle \int_{|z|=4} \frac{\mathrm{d}z}{9+z} $.

but $|9|>4$, how to do the integration in this case ?

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