I did the same thing in about as many steps as chrono. Will post my steps from there.
EDIT: Here are my steps to follow on...
code:
dy/dx = int( (x^4 + 2x^2 + 2 + 2/x^2 + 1/x^4)^1/2 ) what you had
y = (x^4 + 2x^2 + 2 + 2/x^2 + 1/x^4)^3/2 reversal of the chain rule
3/2 * (4x^3 + 4x - 4/x^3 - 4/x^4)
y = (x^4 + 2x^2 + 2 + 2/x^2 + 1/x^4)^3/2 simplified somewhat
6 * (x^3 + x - 1/x^3 - 1/x^4
see why I wanted to simplify it a bit.....