In equality 2

Let $a$ , $b$ , and $c$ be positive reals so that $abc=1$ . The smallest possible value of $\dfrac{a^3}{(b+1)(c+1)}+\dfrac{b^3}{(a+1)(c+1)}+\dfrac{c^3}{(a+1)(b+1)}$ can be expressed as $\frac{A}{B}$ , where $A$ and $B$ are positive integers that are relatively prime. What is $A+B$ ?