The integral is studied in On Certain Indefinite Integrals Involving Bessel Functions (1958).
(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.)
The paper is behind a paywall, so I have not studied it.
And then there is Tables of some indefinite integral of bessel functions of integer order, (2017) which examines the $i=0$ integral and gives both a small-$x$ and a large-$x$ series expansion, see pages 87 and following.