My second derivative is
$$\frac{(250-64 x^3)}{(3 x (-125+8 x^3))^\frac{5}{3}} $$ I know that function is undefined if $x = 0$ and $x=5/2$. What would be concavity intervals? Why not $(-\infty,0)$,$(0,5/2)$ - Increasing and $(5/2,\infty)$ - decreasing ?
The top is negative up to $x=(125/32)^{1/3}\approx 1.575$. The bottom is positive for $x\lt 0$, negative for $0\lt x\lt 5/2$, and positive for $x\gt 5/2$.
Thus for $x\lt 0$, the second derivative is negative, and therefore the first derivative is decreasing, the graph is concave down.
From $0$ to $(125/32)^{1/3}$, the top is still negative. The bottom is negative, so the ratio is positive, the graph of our function is concave up (some people say convex).
From $(125)^{1/3}$, for similar reasons the graph is concave down.
Finally, past $5/2$, the curve is concave up.