Integral of a product of functions divided by the integral of one of the two functions Ratio between two integrals: $$\frac {\int f(x)g(x)} { \int f(x)}.$$ 
Does exist a rule or do you know a way to solve it?
$f(x)= (1+x)^n  e^{-ax}$
$g(x)= \ln(1+x)$
So: the numerator is the integral of $f(x)g(x)$ and the denominator is only the integral of $f(x)$.
And the question is if there exists a solution of this ratio..

My complete problem is: $$\frac {\int f(x)g(x)} {\left( \int f(x)\right)^2}.$$  

In the case of definite integrals, we have the following theorem, called the weighted mean value theorem: If $f$ and $g$ are continuous on a closed interval $[a,b]$, and if $g$ never changes sign in $[a,b]$, then we have 
$$ \int_{a}^{b} f(x) \cdot g(x) \ dx = f(c) \cdot \int_{a}^{b} g(x) \ dx $$ for some $c$ in $[a,b]$. For reference, see Calculus vol. 1 by Tom M. Apostol, Theorem 3.16. Hope this piece of information is of help. 
