I was computing some kind of marginal likelihood and came up with the following improper integral,
$$\int_{-\infty}^{\infty}\frac{e^{kx}}{\prod_{i=1}^{m} (e^x+a_i)^{b_i}}dx$$
Or,
$$\int_0^\infty\frac{w^{k-1}}{\prod_{i=1}^{m}(w+a_i)^{b_i}}dw$$
where $a_i>0$; $b_i, k \in \Bbb{N}$; $0<k<\sum b_i$
It is possible to try to expand the integrand into partial fractions, calculate the integral by part and take the limit. However, I could not find a general representation. It would be of great help if someone could give me some insight on this. Thanks!
Split off the factor $a_i^{b_i}$ in the denominator, the write $(w/a_i+1)^{b_i}$ as its geometric series, and exchange integration and summation. First verify that the integrals converge, though.