I have following function:
$$ f(x)=(x+2)[(a+x)^2Ei(-a-x)\exp(a+x)+a]+x(x+1), $$
where $0<a<1$ and $x>0$ and $Ei(x)$ is the exponential integral. Is it possible to show that $f(x)$ is always positive? I have plotted this function over various values of $a$ in WA and the plot shows that $f(x)$ is positive. So can we prove it analytically? Any help in this regard will be much appreciated. If we can not prove it analytically then is there any other method to prove it.