I have been struggling to solve the following integral. It seems close to a Gamma function, but not quite.
$\int_0^\infty dx x^d e^{-(ax^2+bx+c)}$
My first thought was to complete the square,
$A\int_0^\infty dx x^d e^{-(\sqrt{a}x+B)^2}$,
and make a change of variables.
$u = \sqrt{a}x + B$
However, I get stuck because with the change of variables, the $x^d$ term gets ugly, and the lower limit is no longer zero. For the lower limit not being zero, I can accept that maybe it comes out like an incomplete gamma function, or something like that. As for the binomial that is to the $d$ power after change of variable, maybe expanding it could help, but I'm not sure.
Any wisdom would be gratefully received. I have tried using Wolfram, which suggested the solution should involve confluent hypergeometric functions, but wasn't able to confirm the connection myself. I also carefully read wikipedia and Shaum, but I feel out of luck!