Not just AM-GM

Algebra Level 4

$\frac{1}{a^{3}(b+c)} + \frac{1}{b^{3}(c+a)} + \frac{1}{c^{3}(a+b)} \geq \frac{A}{B}$

The inequality above, where $A$ and $B$ are coprime, holds true for $abc = 1$ and $a, b, c > 0$ . Find $A+B$ .