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Calculus Level 5

\[\sum _{ n=0 }^{ \infty }{ \frac { \psi \left( n+1 \right) +\gamma }{ { \left( n+1 \right) }^{ 4 } } = } A\zeta \left( B \right) -\zeta \left( C \right) \zeta \left( D \right) \]

The equation above holds true for positive integers \(A,B,C\) and \(D\). Find \(A+B+C+D\).

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