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∫0∞∫0∞∫0∞∫0∞1wxyz(w+x+y+z+1w+1x+1y+1z)2 dw dx dy dz\displaystyle \int \limits_0^\infty \int \limits_0^\infty \int \limits_0^\infty \int \limits_0^\infty \frac{1}{wxyz \left(w+x+y+z+\dfrac1w + \dfrac1x + \dfrac1y + \dfrac1z \right)^2} \ dw \ dx \ dy \ dz 0∫∞0∫∞0∫∞0∫∞wxyz(w+x+y+z+w1+x1+y1+z1)21 dw dx dy dz
If the integral above is equal to ABζ(C)\dfrac{A}{B} \zeta(C)BAζ(C), where AAA and BBB are coprime positive integers and CCC is an integer, find A+B+CA+B+CA+B+C.
Notation: ζ(⋅)\zeta(\cdot) ζ(⋅) denotes the Riemann zeta function.
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