Can ${}_2F_1\left(1,b;1+b;z\right)$ be simplified?

Can ${}_2F_1\left(1,b;1+b;z\right)$ be simplified?

I would like to ask a question about hypergeometric function ${}_2F_1$.
Specifically, I hope to solve the equation $$ \frac{\delta
x}{1-\delta}{}_2F_1\left(1,1-\delta;2-\delta;x\right)=a, $$ where
$\delta\in\left(0,1\right)$ and $a\geq 1$. Is the $x$ has the closed-form
solution? If not, for the special case when $\delta=\frac{1}{2}$, can we
get $x$?
I think the above hypergeometric function has the special structure, but I
checked the literatures and cannot find useful results. I appreciate if
you can help to solve it, or give some suggestions.
Thanks.