Playing Carts - I

Consider a Cartesian plane where the functions $f(x) = x^3, \; f'(x)$ and $f''(x)$ are all plotted.

Let $\triangle{\alpha}$ be the triangle with vertexes on all lattice points for which one of the functions intersects another.

Let $\triangle{\beta}$ be the triangle with vertexes on the first quadrant points for which one of the functions intersects another.

Evaluate the approximate numerical value of the sum of $\alpha$ 's area and perimeter plus $\beta$ 's area and perimeter.