Solve $\int \frac{\lambda^{\alpha+\beta}\theta^{-m}}{\prod_{s=1}^n (1/U_s+\lambda)^{\alpha+1}\prod_{s=1}^m (1/(T_s\theta)+\lambda)^{\beta+1}}d\lambda$

Question

I need to compute the numerical value of this integral, hundred thousand of times, for a typical dataset. How can I get a solution or a good approximation for $$\int_0^\infty f(\lambda) d\lambda $$

where

$$f(\lambda)=\frac{\lambda^{\alpha+\beta}\theta^{-m}}{\prod_{s=1}^n (1/U_s+\lambda)^{\alpha_s+1}\prod_{s=1}^m (1/(T_s\theta)+\lambda)^{\beta_s+1}}$$

where $0<U_s<\alpha$, $0<T_s<\beta$, all $\in \Bbb N$, $\theta\in \Bbb R^+$, and

$$\sum_s \alpha_s =\alpha, \sum_s \beta_s =\beta$$