$_2F_1\left(\frac34,\frac54,2,x^2\right)$ in terms of elliptic integrals $E$ and $K$

Question

I am given the following formula (found to be correct numerically): $$ _2F_1\left(\frac34,\frac54,2,x^2\right) = \frac{-8}{\pi x^2} \left[ \sqrt{1+|x|}~E\left(\frac{2|x|}{1+|x|}\right)- \frac{1}{\sqrt{1+|x|}}K\left(\frac{2|x|}{1+|x|}\right) \right] \quad(|x|<1), $$ where $_2F_1$ is the Gauss hypergeometric function, and $E$ and $K$ are the elliptic integrals.

Where can I find a formula or clue in mathematical tables?

Thank you for viewing, and help me please.

Answer 1

This formula is referenced on WolframAlpha : http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/03/09/21/03/0037/

and confirmed :

http://www.wolframalpha.com/input/?i=hypergeometric2F1%283%2F4%2C5%2F4%2C2%2Cx%5E2%29&x=0&y=0