I am trying to obtain a closed form solution of this definite integral, or in a form at least which simplify its numerical treatment. $$\int_{x_1=0}^1...\int_{x_N=0}^1 \prod_r \left( \frac {x_r f_r} {\sum_g x_g f_g}\right) ^{n_r} dx_1...dx_N $$ where $n_r$ are positive integers, and $f_r\ge0$ constants.
I tried to introduce a contour transformation, by $s=x_1 f_1 + ... + x_n f_n, y_2=x_2 f_2,..., y_N=x_N f_N$, |det Jacobian|=constant, which "reduce" to $$\int_{s=0}^{\sum_g f_g}ds\int_{y_2=0}^{?_2}dy_2...\int_{y_N=0}^{?_N}dy_N \prod_r \left( \frac {y_r} s\right) ^{n_r}$$
where $y_r$ are constrained (integral upper limits $?_r$) to total $s$ and be no greater than $f_r$, which seems no advance at all to me .
Your multiple integral $I$ can be converted into a single integral (to start with - more simplifications are perhaps possible). Use the identity $$ x^{-s}=\frac{1}{\Gamma(s)}\int_0^\infty d\xi\ \xi^{s-1}e^{-\xi x} $$ with $s=\sum_r n_r$ to 'exponentiate' the denominator and obtain $$ I = \left(\prod_r f_r^{n_r}\right)\frac{1}{\Gamma(\sum_r n_r)}\int_0^\infty d\xi\ \xi^{\sum_r n_r-1}\prod_{r=1}^N \left[\int_0^1 dx\ x^{n_r}e^{-\xi f_r x}\right] $$ $$ = \left(\prod_r f_r^{n_r}\right)\frac{1}{\Gamma(\sum_r n_r)}\int_0^\infty d\xi\ \xi^{\sum_r n_r-1}\prod_{r=1}^N \left[(f_r \xi )^{-n_r-1}\gamma(n_r+1,\xi f_r)\right]\ $$ $$ =\left(\prod_r\frac{1}{f_r}\right)\frac{1}{\Gamma(\sum_r n_r)}\int_0^\infty d\xi \frac{\prod_{r=1}^N \gamma(n_r+1,\xi f_r)}{\xi^{N+1}}\ , $$ where $\gamma(a,x)=\int_0^x dt\ t^{a-1}e^{-t}$ is the lower incomplete Gamma function http://mathworld.wolfram.com/IncompleteGammaFunction.html
Further simplifications are perhaps possible, noting that the lower incomplete Gamma function whose first argument is an integer (as in this case) has an elementary representation as $(1-\text{a finite sum})$ (see the link above).