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Calculus Level 5

\[\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{A}}{B} \Big[ \psi_{1} \left(\frac{C}{D} \right) - \psi_{1} \left(\frac{E}{F} \right) \Big]- \frac{G}{H} \zeta(I) \]

Here, \(A,B...,I\) are positive integers. Find \(\min(A+B+C+D+E+F+G+H+I)\)

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