Hello,
i have found a nice online article which explains in great detail the
algorithm from Legendre allowing to reduce general elliptic integrals
to standard elliptic integrals of first, second and third form.
http://everything2.com/e2node/elliptic%2520integral%2520standard%2520forms
Here the aim is to arrange so that branch cuts are symmetric with respect to
0, and this is achieved through clever tricks.
--
Michel Talon