Set  up the arc length  integral by hand, then use “Integrate” to evaluate it:

Integrate[ Sqrt[(2*t)^2+(4*t)^2+1], {t,0,2}]

What does this “sinh” function whose inverse appears above look like?

Plot[ Sinh[x], {x,-3,3} ]

Get the  value of the integral as a number, since the symbolic answer isn’t very intuitive.

N[Integrate[ Sqrt[(2*t)^2+(4*t)^2+1], {t,0,2}]]

Finally, notice that Mathematica already has an “ArcLength” function built in that saves you having to set up the integral by hand.

ArcLength[ {t^2+1,2*t^2,t}, {t,0,2}]