HMMT 2005 T8

Suppose $x$ is a fifth root of unity. Find, in radical form, all possible values of

$$2 x+\frac{1}{1+x}+\frac{x}{1+x^{2}}+\frac{x^{2}}{1+x^{3}}+\frac{x^{3}}{1+x^{4}} .$$