I want to find an analytical solution (exact/closed-form) for $x$ of the following minimization problem: $$\min_{x} b x \left[e^\left(\frac{a}{x}\right)-1\right]+d (1-c-x) \left[e^ \left(\frac{a}{1-c-x}\right)-1\right]$$ where $0\leq c+x\leq 1$ and $c,x\geq0$. It can be shown that the above minimization function, say $F(x)$, has a minimum because of $\frac{d^2F(x)}{dx^2}>0$. Therefore, I tried to find $x$ by solving $\frac{dF(x)}{dx}=0$, but it is complicated as I cannot write $x$ in terms of other variables. Here is the expression which I find
$$\frac{dF(x)}{dx}=b \left[e^{a/x} \left(1-\frac{a}{x}\right)-1\right]-\frac{d e^{-\frac{a}{c+x-1}} (a+c+x-1)}{c+x-1}+d=0.$$
Can some one help me to solve this equation for $x$?
Otherwise, can someone suggest me an alternative way to find $x$?