How to evaluate this integral: $$\int_0^1 \int_0^1 \cdots \int_0^1\frac{x_{1}^2+x_{2}^2+\cdots+x_{n}^2}{x_{1}+x_{2}+\cdots+x_{n}}dx_{1}\, dx_{2}\cdots \, dx_{n}=?$$ I'm making use of the integral identity: $\int_{0}^{+\infty }e^{-t(x_{1}+x_{2}\cdots +x_{n})}dt=\frac{1}{x_{1}+x_{2}\cdots +x_{n}}$and then reversing the order of integration with respect to time and space variables. But for $n=1$,then such that,$\int_{0}^{\infty }dt\int_{0}^{1}x^{2}e^{-tx}dx=\int_{0}^{\infty }\frac{2 - e^{-t}(2 + 2t+t^2)}{t^3}dt=\int_{0}^{1}xdx=\frac{1}{2}$,and,$\int_0^1 \int_0^1 \cdots \int_0^1\frac{x_{1}^2+x_{2}^2+\cdots+x_{n}^2}{x_{1}+x_{2}+\cdots+x_{n}}dx_{1}\, dx_{2}\cdots \, dx_{n}=n\int_{0}^{+\infty }\frac{2 - e^{-t}\left ( 2 + 2t+t^2 \right )}{t^3}\left ( \frac{1-e^{-t}}{t} \right )^{n-1}dt$

How to evaluate this integral: $$\int_0^1 \int_0^1 \cdots \int_0^1\frac{x_{1}^2+x_{2}^2+\cdots+x_{n}^2}{x_{1}+x_{2}+\cdots+x_{n}}dx_{1}\, dx_{2}\cdots \, dx_{n}=?$$ I'm making use of the integral identity: $$\int_{0}^{+\infty }e^{-t(x_{1}+x_{2}\cdots +x_{n})}dt=\frac{1}{x_{1}+x_{2}\cdots +x_{n}}$$ and then reversing the order of integration with respect to time and space variables. But for $n=1$, then such that, $$\int_{0}^{\infty }dt\int_{0}^{1}x^{2}e^{-tx}dx=\int_{0}^{\infty }\frac{2 - e^{-t}(2 + 2t+t^2)}{t^3}dt=\int_{0}^{1}x\,dx=\frac{1}{2},$$ and $$\int_0^1 \int_0^1 \cdots \int_0^1\frac{x_{1}^2+x_{2}^2+\cdots+x_{n}^2}{x_{1}+x_{2}+\cdots+x_{n}}dx_{1}\, dx_{2}\cdots \, dx_{n}\\=n\int_{0}^{+\infty }\frac{2 - e^{-t}\left ( 2 + 2t+t^2 \right )}{t^3}\left ( \frac{1-e^{-t}}{t} \right )^{n-1}dt.$$