I would like to solve the following integral that is a variation of this one (Integral involving Modified Bessel Function of the First Kind).
Namely, I have:
$$\frac{1}{\sqrt{2\pi w^2}}\int_{-\infty}^{+\infty} \, x \, e^{(-\frac{x}{\alpha}-\frac{1}{2w^2}(x-\hat{x})^2)} \, I_0\left(\frac{x}{\beta}\right)\,dx .$$
By using the series representation for $I_0(x)$, I obtain the Tricomi hypergeometric function and I don't know how to calculate the corresponding series.
Any suggestion is highly appreciated.