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I am trying to find a closed form for this integral:

$$\int_{0}^{\infty}\int_{0}^{\infty}e^{-d_{p,s}^v\,x-d_{s,p}^v\, y+d_{p,p}^v\,\frac{\xi_1\,\sigma^2\,xy}{P_p\,\xi_2\,xy+\sigma^2}}\mathrm dy\mathrm dx,$$

where $d_{p,s}^v,d_{s,p}^v,d_{p,p}^v,P_p,\xi_1,\xi_2,\sigma^2$ are positive constant.

I have got the following expression with preliminary calculation for the above integral:

$$\frac{e^{d_{p,p}^v}\frac{\xi_1\sigma^2}{P_p\xi_2}}{d_{s,p}^v}\int_{0}^{\infty} e^{-d_{p,s}^v\,x+\frac{d_{s,p}^v\,\sigma^2}{P_p\,\xi_2\,x}}\,*\gamma(1,\frac{d_{s,p}^v\,\sigma^2}{P_p\,\xi_2\, x};\frac{d_{s,p}^v\,\xi_1\,\sigma^4}{P_p^2\xi_2^2x})\mathrm dx,$$

where $\gamma(a,t;b)$ is defined as $$\gamma(a,t;b)=\int_t^\infty{x^{a-1}e^{-x-\frac{b}{x}}}\mathrm dx$$ in some references.

I'm wondering if there is a closed form for the integral. Thank you for your answer!

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