Wait a second, I just had some inspiration. I'll let you know If I truley need help...
I tried get a tidy image of the integral with Integrator, but it told me the answer does not exist. I'm almost certain it does however.
(1+ln(x))*sqrt(1+(xln(x))^2)
Help! I tried integration by parts but it turns into a jumbled mess, unless I'm overlooking something.