Old topic, but without really satisfactory answers, in my opinion.
I'd go with multline, which has been introduced exactly for cases such as the present one. On the other hand, also split may be considered worth of use in this case, particularly in the third example, where the alignment is on the right, underlining the similarity between the two parts.
\documentclass{article}
\usepackage{amsmath}
\usepackage{leftidx}
\newcommand{\hg}[3][F]{\mathop{\leftidx{{#2}}{#1}{{#3}}\kern-\scriptspace}}
\begin{document}
\subsection*{\texttt{multline}}
\begin{multline}
A_0 =
\frac{1}{(\alpha+t_x)^{r+s+x}} \hg{2}{1} \biggl(
r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + t_x}
\biggr) \
-\frac{1}{(\alpha+T)^{r+s+x}}{} \hg{2}{1} \biggl(
r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + T}
\biggr),
\end{multline}
\subsection*{\texttt{split} 1}
\begin{equation}
\begin{split}
A_0 &=
\frac{1}{(\alpha+t_x)^{r+s+x}} \hg{2}{1} \biggl(
r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + t_x}
\biggr) \
&\qquad
-\frac{1}{(\alpha+T)^{r+s+x}}{} \hg{2}{1} \biggl(
r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + T}
\biggr),
\end{split}
\end{equation}
\subsection*{\texttt{split} 2}
\begin{equation}
\begin{split}
A_0 &=
\frac{1}{(\alpha+t_x)^{r+s+x}} \hg{2}{1} \biggl(
r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + t_x}
\biggr) \
&
-\frac{1}{(\alpha+T)^{r+s+x}}{} \hg{2}{1} \biggl(
r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + T}
\biggr),
\end{split}
\end{equation}
\subsection*{\texttt{split} 3}
\begin{equation}
\begin{split}
A_0 =
\frac{1}{(\alpha+t_x)^{r+s+x}} \hg{2}{1} \biggl(
r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + t_x}
\biggr) \
-\frac{1}{(\alpha+T)^{r+s+x}}{} \hg{2}{1} \biggl(
r+s+x,x+1;r+s+x+1;\frac{\alpha-\beta}{\alpha + T}
\biggr)\makebox[0pt][l]{,}
\end{split}
\end{equation}
\end{document}
Points to note:
instead of {}_2F_1 that produces misaligned subscripts, I defined a command \hg (for hypergeometric);
\left and \right have been replaced by \biggl and \biggr;
if no numbering is wanted just add * to the name of the outer environment (multline* or equation*).

splitmight be fruitful here, particularly with numbered equations. Of course,multlinedis not needed, asmultline*suffices. – egreg Aug 01 '14 at 09:34equation*,align*, etc -- do not create equation numbers. Use the "non-starred" environments --equation,align, etc -- to create numbered equations. – Mico May 22 '23 at 05:50splitmight be more syntactically fitting, given that there is only one equation here? As The LaTeX Companion, 3ed, puts it, "When a simple alignment is needed within a single multiple-line equation, thesplitenvironment is almost always the best choice." – murray Nov 30 '23 at 15:55