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 $\frac16(B(x,y,z)+\cdots+B(z,y,x))$.

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    $\begingroup$ Seems like some flavor of Selberg's integral could help? $\endgroup$ – Suvrit Aug 21 '17 at 16:49

In view of the analogy with the relation between Gauss sums and Jacobi sums, one should have $$\int_{0\le x,y,z\le1, x+y+z=1}(1-x)^{a-1}(1-y)^{b-1}(1-z)^{c-1}dxdy=B(a,b,c),$$ and similarly with more variables (I didn't check). Of course this is double integral, not a one-variable one as you ask.

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    $\begingroup$ This seems indeed a well motivated guess to directly generalize the beta integral, but as you say, my question is if a simple integral is possible. $\endgroup$ – Wolfgang Aug 22 '17 at 16:18

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