Calculating a multiple convolution with variables bounded both individualy and by total

Question

I am trying to find a closed form or a transformation which simplify the numerical treatment of this multiple integral $$\int_0^{U_1} \cdots \int_0^{U_N} \delta(U,\sum_g u_g) \prod_g u_g^{n_g} \, du_1 \cdots du_N$$ where $n_g$ are non-negative integers, $g=1,\ldots,N$, $\delta(\cdot)$ is the Dirac delta function, $0 \le u_g \le U_g$, $0 < U \le \sum_g U_g$ and $U_i \ne U_j$ in general. Hence, $u_g$ is limited by $U_g$ and are restricted to total $U$.