You sure the question wasn't (x + 1/x)(x^2 + 2 + 1/(x^2))^(1/2) ?
if that was the case then it would be simply the integral of (x + 1/x)^2 but without that +s under the quareroot it makes it almightily difficult. Or even if (x + 1/x) was replaced with the derivative of (x^2 + 1/(x^2)). I'll keep trying my best to get an answer, I got something at home but it's a bit ugly and I have to check with it's derivative just to make sure I did it right but I didn't want to just yet because of it's ugliness