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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 don't know if it provides much help, but at least it is an entry point to the literature (it includes several references where integrals of this type appeared).