On the integral of a function that look like inverse gamma function?

by Arun M   Last Updated March 12, 2017 14:19 PM

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?

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