\[ \large \int_0^\infty \dfrac{\gamma x + \ln [\Gamma(1 + x)]}{x^{9/4}} \, dx = \dfrac{A\sqrt B \pi}C \zeta \left( \dfrac DE \right) \]

The equation above holds true for positive integers \(A,B,C,D\) and \(E\), with \(A,C\) and \(D,E\) coprime pairs, and \(B\) square-free.

Evaluate \(A+B+C+D+E\).

\[\] **Notations**:

\(\gamma \approx 0.5772 \) denotes the Euler-Mascheroni constant.

\( \Gamma(\cdot) \) denotes the Gamma function.

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

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