Approximating Elliptic Integrals on the computer

Raging Dragon

1. The problem statement, all variables and given/known data

Express (EllipticK(x)/EllipticE(x)-1) to two terms with a taylor series

2. Relevant equations

EllipticE(x) = Int((1-k^2*_alpha1^2)^(1/2)/(1-_alpha1^2)^(1/2),_alpha1 = 0 .. 1), `with no restrictions on `(x)

EllipticK(k) = Int(1/((1-_alpha1^2)^(1/2)*(1-k^2*_alpha1^2)^(1/2)),_alpha1 = 0 .. 1)

Note this is the maple code expression for these

3. The attempt at a solution

Using Maple (canadian mathematica) I got -x-5/4*x^3

I mainly want to verify this to make sure I'm solving a particular DE correctly.