If $x^5=2$, find $\frac{x}{x^2+1}+\frac{x^2}{x^4+1}+\frac{x^3}{x+1}+\frac{x^4}{x^3+1}$ 
If $x^5=2$, find
$$\frac{x}{x^2+1}+\frac{x^2}{x^4+1}+\frac{x^3}{x+1}+\frac{x^4}{x^3+1}$$

My attempt-
Since $x^5=2,x^6=2x,x^7=2x^2..$and so on
The equation is equivalent to
$$x^5\left(\frac{1}{2x+x^4}+\frac{1}{2x^2+x^3}+\frac{1}{x^3+x^2}+\frac{1}{x^4+x}\right)$$
Which simplifies to
$$2\left[\frac{3x^4+7x^3+5x^2+8x+12}{2x^4+6x^3+4x^2+6x+12}\right]$$
Now I am struck here.
Any help will be appreciated.
Note: Please don't use root tables and complex numbers too!

$$
y=\frac{x}{x^2+1}+\frac{x^2}{x^4+1}+\frac{x^3}{x+1}+\frac{x^4}{x^3+1}$$
$$
=\frac{x (x^4 + x^3 + x^2 + 1) (x^6 - x^5 + 2 x^4 + x + 1)}{(x + 1) (x^2 + 1) (x^2 - x + 1) (x^4 + 1)}$$
$$
=\frac{8 + 5 2^{1/5} + 7 2^{2/5} + 3 2^{3/5} + 6 2^{4/5}}{3 + 2 2^{1/5} + 3 2^{2/5} + 2^{3/5} + 3 2^{4/5}}$$
$$x=\sqrt[\large 5]{2}\approx 1.1487  \implies y\approx 2.37411
$$
