There is no analytical solution for the complete integral:
sqrt (1 – 1 / (1 + b exp (x / a)) + c / x squared – E)
but it could be approximated, or fitted with a three or four variable least mean squares routine, or integrated numerically. It looks approximately like a bell shaped curve or gaussian. We can take plenty of time with this problem, because it is so important. We have hundreds of pairs of eyes looking at us as we work (in humour).