Hot Integral - 28

Calculus Level 5

$\displaystyle \lim_{n \to \infty}\frac{1}{\Gamma(n+1)} \int\limits_{0}^{\infty} \int\limits_{0}^{\infty} \int\limits_{0}^{\infty} \dots \int\limits_{0}^{\infty} \frac{1}{(\prod_{i=1}^{n} x_i)(\sum_{i=1}^{n} x_i +\frac{1}{x_i})^2}\, dx_1 dx_2 \cdots dx_n$

The above expression can be expressed as $\large \frac{A}{Be^{C\gamma}}$

Write the answer as the concatenation of the sum $$ABC+CBA$$ of the integers $$A,B,C$$. For example, if you think that $$A=1,B=2,C=3$$, type the answer as $$123+321=\boxed{444}$$.

Moreover , if you think that the above expression is not possible and you think that limit does not exist, enter your answer as 888.

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