A Massive Sum
Algebra · problem by Rob C.
Compute the exact value of:$$\sum_{z=0}^\infty\sum_{y=0}^\infty\sum_{x=0}^\infty\frac{x^3+y^3+z^3-3xyz}{3^x \left(3^x3^y+3^y3^z+3^x3^z\right)}$$The answer is in the form $\frac{\alpha}{\beta}$ where $\alpha$ and $\beta$ are coprime integers. Input the value of $\alpha+\beta$