This is a skeleton solution.

By substituting u=(x-2)/sqrt(2) and considering f(x)+f(-x), the integral can be re-written as

A tangent substitution will turn it into a format that Wolfram can solve...finally

https://www.wolframalpha.com/input/?i=integrate+sqrt(1+tan^4+x)+/+(1-tan^2+x)
I know Wolfram used hyperbolic tangent substitution but it is also solvable in MX2 by secant substitution.

Alternatively, if you don't mind handling improper integral, you can do some algebraic manipulation to get:

Substituting v=u

^{-1}+u and w=u

^{-1}-u will lead to two improper (but solvable) integrals because u

^{-1} blows up at 0.