\[ \large \displaystyle \int_0 ^ \infty e^{-x^2} \ln(x)\ \mathrm{d}x = -\dfrac{A}{B}\left( \gamma^C +D\ln(D) \right)\pi^E \]

If the above equation is true for positive integers \(A,B,C,D\), where \(A,B\) are coprime to each other and \(D\) isn't any \(m^{th}\) power of a positive integer with \(m \in \mathbb Z,\ m \geq 2\).

Submit the value of \(A+B+C+D+2E\) as your answer.

**Note:** \( \gamma\) is the Euler-Mascheroni Constant defined as:

\[ \large \gamma =\lim_{n \rightarrow \infty} \left(\displaystyle \sum_{k=1} ^{n} \dfrac{1}{k} -\ln(n) \right) \]

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