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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. $$ {c\;\mbox{$_2$F$_1$}(b,a-1;\,c;\,x)} + \left( x-1 \right) c\;{\mbox{$_2$F$_1$}(a,b;\,c;\,x)} + \left( b-c \right) x\;{\mbox{$_2$F$_1$}(a,b;\,c+1;\,x)} =0 $$

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. $$ {c\;\mbox{$_2$F$_1$}(a-1,b;\,c;\,x)} + \left( x-1 \right) c\;{\mbox{$_2$F$_1$}(a,b;\,c;\,x)} + \left( b-c \right) x\;{\mbox{$_2$F$_1$}(a,b;\,c+1;\,x)} =0 $$

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.

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. $$ {c\;\mbox{$_2$F$_1$}(b,a-1;\,c;\,x)} + \left( x-1 \right) c\;{\mbox{$_2$F$_1$}(a,b;\,c;\,x)} + \left( b-c \right) x\;{\mbox{$_2$F$_1$}(a,b;\,c+1;\,x)} =0 $$

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.

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.