Mutant Ellipsoid

Calculus Level 5

The volume of the region bounded by \(16x^{2}+81y^{4}+64z^{6}=1\) can be expressed, in simplest terms, as \({\Large \frac{\Gamma \left( \frac{A}{B} \right)\Gamma \left( \frac{C}{D} \right)}{E\times \Gamma \left( \frac{F}{G} \right)}\sqrt{\pi }}\), where \(A, B, C, D, E, F\) and \(G\) are positive integers, with \(\gcd(A,B) = \gcd(C,D) = \gcd(F,G) = 1\).

If we know that \(\dfrac AB, \dfrac CD , \dfrac FG < 1 \), find \(A+B+C+D+E+F+G\).


Thanks to Michael Mendrin for finding the simplification error in the original problem.
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