Closed-form solution for $x^a = (1-x)^{1-a}\cdot b$ with $0 < a < 1$

Question

How can I tell if $x^a = (1-x)^{1-a} \cdot b$ has a closed form solution for $x$, assuming $0<a<1$. It does in the case of $a=\frac{1}{2}$, but is this the only case?

Computer algebra systems don't seem to be helping.

Thanks in advance for any tips.

Answer 1

Maple also finds closed form solutions for $a = \frac 1 3, \frac 1 4, \frac 2 3, \frac 3 4, \frac 2 5, \frac 3 5$...

Wolfram Alpha online also get them: cf for $a=\frac 13$ and for $a = \frac 1 4$.

The 2 shortest answers: