How to compute inner product in the Hardy space

[LikeMath]

Hi,
Let H^2 be the Hardy space on the open unit disk.
I am wondering how can I compute the following inner product

<\frac{1}{\left(1-\overline{\alpha_1} z\right)^2}\frac{z-\alpha_2}{1-\overline{\alpha_2} z},\frac{z}{\left(1-\overline{\alpha_1} z\right)^2}>,

where \alpha_1,\alpha_2 in the unit disk.

I tried to expand the functions but it became complicated. Also it did not work with the integration.

Is there an idea to be tried?

Thanks in advanced
Likemath

 

[LikeMath]

Any idea?