The integral is studied in <A HREF="http://onlinelibrary.wiley.com/doi/10.1002/sapm1958371157/abstract">On Certain Indefinite Integrals Involving Bessel Functions</A> (1958).    
<sub>(The $i=0$ integral is $g(a,0,x)$ in the notation of that paper, and as Robert Israel points out, the $i=1$ integral is simply related.)</sub>   
The paper is behind a paywall, so I have not studied it.

And then there is <A HREF="http://web.eah-jena.de/~rsh/forschung/stoer/besint.pdf">Tables of some indefinite integral of Bessel functions of integer order</A> (2017), which examines the $i=0$ integral in great detail and gives several series expansions, see pages 87 and following.

I reproduce one such formula:

<IMG SRC="https://ilorentz.org/beenakker/MO/Bessel_series_5.png"/>