Integral Involving Gaussian Q function and exponential

Question

I have a question regarding an integral that involves the Gaussian Q-function and exponential.

The integral has the form

$f(x)= \int_{0}^{\infty}{x^a e^{(-b^2 x^2)} Q(c - x) dx}$

where $a$, $b$ and $c$ are constants. $Q(.)$ is the gaussian Q function. I was wondering if there is a closed form for this integral or it is included in the table of integral under some similar form.

Thanks in advance

Answer 1

The paper by Y. Chen and N.C. Beaulieu, "Solutions to Infinite Integrals of Gaussian Q-Function Products and Some Applications," in IEEE Communications Letters discusses the integrals of the form

$\displaystyle \int_{0}^{\infty}x^{a}e^{-bx^{2}}Q(cx) dx$.

May be it can help.