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Someone has a reference that addresses an integral of the followns type $$I_{a,b,x} = \int_{0}^{+\infty} \zeta^{-a} \, \Gamma\left(\zeta+b\right) \, W_{-\zeta-b,\tfrac{-1}{2}}(x) \, d\zeta$$ where $\Gamma(.)$ is Euler's Gamma-function and $W_{\kappa,\mu}(x)$ is the Whittaker function, with $a,b>0$ and $x\in \mathbb R$.

Thanks you in advance

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