Integral of Bessel function of the first kind and exponential function

Question

I would need to know if there's a closed form for the following integral:

$$\int_{0}^{\infty} x^{-1}J_{\frac{1}{2}}(\pi x)J_{\frac{1}{2}}(\pi x)\exp(-b(x-x_0)^2)$$

with $b>0$ and $x_0\in \mathbf{R}$

I know that there is a closed form for $x_0=0$, but I need the general case. Thanks