While working on mixture (variance) of normal distribution and keep running into these two integrals
$$ \int_0^{\infty}\dfrac{v}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v}}dv,$$
$$\int_0^{\infty}\dfrac{v^{-1}}{\sqrt{v + c}}e^{-\dfrac{y^2}{2(v + c)} - \dfrac{(u-v)^2}{u^2v} }dv,$$
where $c>0, u>0 ,y\in \mathbb R$.
I was wondering are they solvable? Can they be expressed as some known function or in elementary terms?
Any help would be appreciated.