\[\large \int_2^3 \dfrac{2x^5 + x^4 - 2x^3 + 2x^2 + 1}{x^6 + x^4-x^2 - 1} \, dx \]

If the integral above can be expressed in the form of

\[ \dfrac AB \ln 6 - \dfrac DE \; , \]

where \(A,B,D\) and \(E\) are positive integers with \( \gcd(A,B) = \gcd(D,E) = 1\), find \(6\times A \times B \times D \times E \).

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