I've officially given up on being able to do this on my own. I keep making mistakes somewhere, and can't find where, so I'm throwing it to you guys.
Arc Length of a curve from -27 to 512, the integral, which I know is right is:
sqrt(1+((4 y^(1/3)-1/(16 y^(1/3)))^2)) from -27 to 512
It's pretty much a terrifying experience trying this one.