Find the smallest value of $\sqrt[5]{\frac{abc}{b+c}} + \sqrt[5]{\frac{b}{c(1+ab)}} + \sqrt[5]{\frac{c}{b(1+ac)}}$ Let $a\ge0$ and $b,c>0$, we need to find the smallest value of the expression 
$$S=\sqrt[5]{\frac{abc}{b+c}} + \sqrt[5]{\frac{b}{c(1+ab)}} + \sqrt[5]{\frac{c}{b(1+ac)}}$$
I have no idea for this question, does anyone can help me to answer this?

For $a=0$ and $b=c=1$ we have a value $2$.
We'll prove that it's a minimal value.
Indeed, let $a\neq0$, $a=x$, $b=\frac{1}{y}$ and $c=\frac{1}{z}$.
Hence, by AM-GM we obtain: $$S=\sum_{cyc}\sqrt[5]{\frac{x}{y+z}}\geq\sum_{cyc}\sqrt{\frac{x}{y+z}}=$$
$$=\sum_{cyc}\frac{2x}{2\sqrt{x(y+z)}}\geq\sum_{cyc}\frac{2x}{x+y+z}=2$$
and we are done!
