The beta function $B(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}$ can be written for $\Re x, \Re y > 0$ symmetrically as $$ B(x,y) = \int_{-\frac12}^{\frac12}(\frac12-t)^{x-1}(\frac12+t)^{y-1}\,dt.$$
Let us define more generally $B(x,y,z):=\dfrac{\Gamma(x)\,\Gamma(y)\,\Gamma(z)}{\Gamma(x+y+z)}=B(x,y)\cdot B(x+y,z)$

and likewise for more variables.

Is it possible to write $B(x,y,z)$ in a similar way as a simple (i.e. one-variable) integral?

I guess it would be asking too much if requiring moreover the integrand to be (more or less) symmetric in $x,y,z$. That is, other than a trivial mean $\frac16(B(x,y,z)+\cdots+B(z,y,x))$.