Concise Solution for Algebra Problem Consider the following expression 
$$\sum_{i=1}^n \left(\frac{p\alpha_ie^{\alpha_i\cdot x}}{1-p+pe^{\alpha_i\cdot x}}\right) = c$$
where $\alpha_i, c \in \mathbb{R}$ and $p \in (0,1).$
How do I solve for $x$ from the equation?

Here's what I tried and it was too long to be a comment.
By integrating both sides:
$$\sum_{i=1}^n\ln(1-p+p\;e^{\alpha_i x})=cx+c_1$$
and since this must hold for $x=0$, then $c_1=0$.
Therefore
$$\prod_{i=1}^n\left(1+p(e^{\alpha_i x}-1)\right)=e^{cx}$$
Set $y:=e^x$ to convert the above equation to a nonlinear (and seemingly unsolvable in general) equation:
$$\prod_{i=1}^n\left(1+p(y^{\alpha_i}-1)\right)=y^{c}$$
