integration - how can I evaluate this integral?

how to evaluate this integral:
$$l(y)=\int\limits_\beta^\infty \theta\exp(-y\theta)\alpha\exp(-\alpha\theta) \, d\theta$$
where $\alpha,\beta,\theta,y>0.$
Because I find it infinity!
Can anyone help me to evaluate this integral? Thank you.$$$$
I find this solution :$$\left(\left.\frac{-1}{(\alpha+y)^2}\exp(-(\alpha+y)\theta)\right)\right|_{\beta}^\infty-\left.\frac{\theta}{(\alpha+y)}\exp(-(\alpha+y)\theta)\right|_{\beta}^\infty.$$ $$$$
in which in the second term, I obtain infinity value!