Integration of a square root

While working on finding the arc length of the parametrized curve $\displaystyle g(t)=(logt,2t,t^2)$ where $\displaystyle t$ is in $\displaystyle [1,e]$. i got to an integral that i couldn't manage to integrate. Here is the work so far leading up to the problematic integral so that it may be checked for errors.

arc length $\displaystyle =\int_{a}^{b} |g'(t)| dt $ by definition

$\displaystyle =\int_{1}^{e} |( 1/t,2,2t)| dt $ by taking the partials of g

$\displaystyle =\int_{1}^{e} \sqrt{(1/t)^2+2^2+(2t)^2} dt $ by taking the length

$\displaystyle =\int_{1}^{e} \sqrt{1/t^2+4+4t^2} dt $ by simplification

$\displaystyle =$ ...? by integration

I'm thinking it has something to do with simplifying the inside of the sqrt some more, or maybe a trig substitution (ugh!)

By the way, the answer is $\displaystyle e^2$

(Appologies if this is posted in the wrong thread as it is coming from an analysis course, but it really is only the integral that i am needing help with, so i see it fit that i place this thread here. Thank you)