Solve a definite integral with a gaussian integrand.

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How do I solve the following integration: $$\int_{0}^{\infty}dx\:\left(\frac{1}{x+\sqrt{x^{2}-b^{2}}}\right)^{c}\frac{(x_{0}-x)\, e^{-\alpha\,(x_{0}-x)^{2}}}{\sqrt{x^{2}-b^{2}}}$$ Here, $\alpha,\, b>0$