Not just AM-GM

Algebra Level 4

1a3(b+c)+1b3(c+a)+1c3(a+b)AB\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 AA and BB are coprime, holds true for abc=1abc = 1 and a,b,c>0a, b, c > 0. Find A+BA+B.

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