I am trying to calculate the following integral
$$ I = \int \mathrm{d} x \ x \ J_0(ax) J_0(bx) J_1(cx) $$
which can be thought of as a particular case of the more general integral
$$ I(n_1,n_2,n_3) = \int \mathrm{d} x \ x \ J_{n_1}(ax) J_{n_2} (bx) J_{n_3} (cx) $$
I am aware of the analytic result for I(0,0,0), for I(0,1,1) and also for general $I(n_1,n_2,n_3)$ when $n_1 + n_2 + n_3 = 0$, however I cannot find a result for $I=I(0,0,1)$.
So far I have tried expressing $J_1(cx)$ using recursion relations to try and recover other known integral from $I$, or integrating by parts. Both these approaches seems to make the integral more complicated.