# Evaluting $\int_0^{\infty}\frac{v}{\sqrt{v + c}}e^{-\frac{y^2}{2(v + c)} - \frac{(u-v)^2}{u^2v}}dv$

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.

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