How to evaluate the following integral? $$\int_{0}^{\infty} x^{-\alpha-1} \exp\left(-\sum_{i=1}^{n} \frac{\gamma_i}{x+\beta_i}\right)\mathrm{d}x~,$$ where the parameters $\alpha, \gamma_i, \beta_i$ $> 0$ where $1 \leq i \leq n$.
For $n = 1$ and $\beta_1 = 0$ the integrand resembles an inverse gamma function, which can be evaluated with the help of an inverse gamma distribution. However, when there is a scalar added to $x$ and there are a few terms summed as shown here, this becomes a bit complicated. How can this be evaluated analytically?