Maple does it in terms of complete elliptic integrals $\rm{K}$ and $\rm{E}$ ...
$$
{\mbox{$_2$F$_1$}\left(\frac12,\frac12;\,2;\,t\right)}={\frac {4\left( t-1 \right){\rm K} \left( 
\sqrt {t} \right)   +4{\rm E} \left( 
\sqrt {t} \right) }{t\pi}}
$$
But that does not show it is **elementary**.  In fact, I suspect it is not elementary.