Definite integral of product of Bessel function, exponential function and power function.

Question

I am attempting to compute the integral $$\int_a^\infty e^{-kr}r^{\alpha-1}J_\nu(\beta r)dr$$ where $J_\nu$ is a bessel function of the first kind of some positive (not necessarily integer) order $\nu$. I've found the identity $$\int_0^\infty e^{-kr}r^{\alpha-1}J_\nu(\beta r)dr=\frac{(\frac{\beta}{2k})^\nu\Gamma(\nu+\alpha)}{k^\alpha\Gamma(\nu+1)}{}_2F_1(\frac{\nu+\alpha}{2},\frac{\nu+\alpha+1}{2};\nu+1;-\frac{\beta^2}{k^2})$$ in Gradshteyn & Ryzhik's table of integrals. However I'd quite like to perform the integral instead from some lower boundary $a>0$, though I am entirely unsure as to how I would go about obtaining this result. Assuming the result even exists in closed form. Any help here would be greatly appreciated.