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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