Just Some Twos!

Say $f(x),g(x),h(x)$ are functions defined for all positive reals $x$ , by the following statements.

- $f(x)=g(x)-h(x)$

- $f(x)=\frac{(g(x))^2-1}{g(x)}$

- $g(x)=x^3$

In the first quadrant, the area enclosed between the three functions and the x-axis is given by $\frac{2^a-2^b-2^c}{2^2}$ . Find $a+b+c$