Maple does it in terms of complete elliptic integrals $\rm{K}$ and $\rm{E}$ ...
$$
{\mbox{$_2$F$_1$}\left(\frac12,\frac12;\,2;\,t\right)}={\frac {4\left( t-1 \right){\rm K} \left( 
\sqrt {t} \right)   +4{\rm E} \left( 
\sqrt {t} \right) }{t\pi}}
\tag1$$
But that does not show it is **elementary**.  In fact, I suspect it is not elementary.  

----

Recall the known formulas:
$$
\rm{K}\big(\sqrt{t}\big) = \frac{\pi}{2}\;
{}_2F_1\left(\frac12 , \frac12 ; 1 ; t\right) ,
\\
\rm{E}\big(\sqrt{t}\big) = \frac{\pi}{2}\;
{}_2F_1\left(-\frac12 , \frac12 ; 1 ; t\right) .
$$
By themselves, they are not elementary.  $(1)$ should follow from these two and a
contiguous formula for the hypergeometrics.
$$
{c\;\mbox{$_2$F$_1$}(a-1,b;\,c;\,x)}
 + \left( x-1 \right) c\;{\mbox{$_2$F$_1$}(a,b;\,c;\,x)}
 + \left( b-c \right) x\;{\mbox{$_2$F$_1$}(a,b;\,c+1;\,x)}
=0
$$