I checked for integrals of the above mentioned form in Gradshetyn and Ryzhik, table of integrals, series and products and Luke's integrals of Bessel functions but couldn't find any. However, I was able to evaluate the integral using Mathematica's NIntegrate and plotted the answer as a function of $x$ but I am seeking to obtain the explicit form of $f(x)$. A guidance on how to go about evaluating this would be appreciated.

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    $\begingroup$ First, reducing the number of parameters is always the best start: $$\int_{-\infty}^{\infty}\frac{e^{-a(x+z)^2}J_{1}(p\sqrt{b^2-z^2})}{\sqrt{b^2-z^2}}dz=\int_{-\infty}^{\infty}\frac{e^{-A(X+t)^2}J_{1}(P \sqrt{1-t^2})}{\sqrt{1-t^2}}dt$$ where $t=z/b$ and other parameters are renamed in obvious way $\endgroup$ – Yuriy S May 29 '17 at 14:31
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    $\begingroup$ I wouldn't hope for an exact value in terms of known special functions though. I'll try and look at it in detail later $\endgroup$ – Yuriy S May 29 '17 at 14:33

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