Arc Length of Ellipse, Hard Integral

[Swimmingly!]

Homework Statement

Find the arc length of the ellipse or deformed circle. r^2=x^2+(y/β)^2

r=radius
β=dilation constant
k="random" constant

Homework Equations

The Attempt at a Solution

I suspect it's impossible but I can't prove that.
After working it out I got stuck with these integrals which I can not solve:

 

[OldEngr63]

Check out the term elliptic integrals. You will find a fully developed theory behind this whole area.

 

[Swimmingly!]

Check out the term elliptic integrals. You will find a fully developed theory behind this whole area.

Thanks. I had googled some pages that hinted at that but this statement in wiki about "elliptic integrals" confirmed it: "In general, elliptic integrals cannot be expressed in terms of elementary functions."