Caner: The complete elliptic integral of the second kind can be expressed as a power series

{\displaystyle E(k)={\frac {\pi }{2}}\sum _{n=0}^{\infty }\left({\frac {(2n)!}{2^{2n}\left(n!\right)^{2}}}\right)^{2}{\frac {k^{2n}}{1-2n}},}.

Or also the complete elliptic integral of the second kind can be expressed

in terms of the Gauss hypergeometric function {}_{2}F_{1}.

for more details see link

https://en.wikipedia.org/wiki/Elliptic_integral

You may find more Links on net

best luck

See alos

http://functions.wolfram.com/introductions/PDF/EllipticE.pdf

Hello Maged ;

This is incomplete Elliptic integral of second kind.Maybe I forgot to mention it.