Calculate: $$I = \int_{\substack{x_1 \geq 0, \ldots, x_n \geq 0 \\ x_1+\cdots+x_n \leq 2}} \sqrt[n]{x_1 \cdots x_n\left(x_1+\cdots+x_n\right)}\, d x_1 \cdots d x_n$$
Attempt: Perform the substitution $ x_i = 2y_i $, the Jacobian will be $ 2^n $ and so $$I=2^{n+1+\frac{1}{n}} \cdot \int_{\substack{x_1 \geq 0, \ldots, x_n \geq 0 \\ x_1+\cdots+x_n \leq 1}} \prod x_i^{\frac{1}{n}} \cdot \sqrt[n]{\left(\sum_j x_j\right)}\,dx_i$$
I don't know how to continue, any ideas?
Thanks for the help!