Summing the sum

Calculus Level 5

\[\large \sum _{ a=1 }^{ \infty }{ \sum _{ b=1 }^{ \infty }{ \dfrac { 1 }{ { a }^{ 2 }{ ( a+b ) }^{ 4 } } } } =-\dfrac { A }{ B } \zeta \left( C \right) +( { \zeta }( D ))^E\]

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

Notation: \(\zeta(\cdot) \) denotes the Riemann zeta function.


Inspiration.

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