Why don’t you try the analytic method for cubics? The Abel-Ruffini only says that the anti-derivative is not elementary, not that it doesn’t exist. The eliptic integrals are non-elementary, but calculating them on a computer is not really different in principle than e.g. computing natural logarithm.
If you plug the desired integral into Mathematica or Wolfram Alpha, it should just spit out the relevant anti derivative expressed in terms of some standard elliptic function, which can then be calculated using some series expansion (or some library of math functions).
If you plug the desired integral into Mathematica or Wolfram Alpha, it should just spit out the relevant anti derivative expressed in terms of some standard elliptic function, which can then be calculated using some series expansion (or some library of math functions).