Integral of the modified Bessel function involving power and exponential functions

Is there a closed-form expression for an integral of the modified Bessel function of the first kind zero-order including the following? $$ \int_{0}^{\infty}x^{-1}e^{-ax}I_{0}(bx^{1/2})dx. $$

where a and b are positive real.

Thanks in advance for both your time and patience.

Solution 1:

This integral doesn't converge. An exactly similar integral formula is provided by Gradshteyn and Ryzhik, but it would be aberrant to use it, as the condition $\Re\left(\nu+\mu+\dfrac{1}{2}\right) > 0$ isn't satisfied in your integral.

$$\int_0^\infty x^{\mu-\frac{1}{2}}e^{-\alpha x}I_{2\nu }\left(2\beta\sqrt{x}\right)\mathrm dx=\frac{\Gamma\left(\mu+\nu+\frac{1}{2}\right)}{\Gamma(2\nu+1)}\beta^{-1}e^{\frac{\beta^2}{2\alpha}}\alpha^{-\mu}M_{-\mu,\nu}\left(\frac{\beta^2}{\alpha}\right) \\ \left[\operatorname{Re}\left(\mu+\nu+\frac{1}{2}\right)>0\right]$$