Everyone. I am trying to solve the following integral: $$ \int_{0}^{\infty}{x \over 1 + a\,x^{-\alpha}}\, \exp\left(-\,{\left[\,x + c\,\right]^{2} \over 2b}\right)\,\mathrm{d}x\quad \mbox{where}\quad a, b, c, \alpha > 0. $$ This integral comes from one problem related to stochastic geometry.
And I've tried $\texttt{Mathematica}$ and $\texttt{Matlab}$, and none of them are able to give the symbolic results, which make me think it is impossible to obtain the results directly. Therefore, my purpose to ask the question is to know whether there exists any kind of approximations of the integral or not or some tricks to solve the integral. Thank you in advance.
I am pretty sure that you can use the Laplace approximation for this integral:
https://en.wikipedia.org/wiki/Laplace%27s_method
$\frac{x}{a + x^{-\alpha}}\exp(-\frac{[x+c]^{2}}{2b}) = \exp(\ln(\frac{x}{a + x^{-\alpha}})-\frac{[x+c]^{2}}{2b})$
Now for the Laplace approximation, consider the function $f(x) = \ln(\frac{x}{a + x^{-\alpha}})-\frac{[x+c]^{2}}{2b}$