I am trying to calculate this integral. I know it has an analytic expression when $a = 0$. But, is there any analytic expression for this case?
Thanks in advance.
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Using formula (18.17) at this link here, you can get a power-series for $J_2(bx)J_1(cx)$, which you can simplify and integrate term-by-term to obtain a "closed-form" expression for your integral. Maple or Mathematica might be able to simplify that even further.
Alternatively, you can look the book Integrals of Bessel Functions by Y. L. Luke, McGraw Hill. 1962.