\[\large{\int_0^1 \dfrac{x^3}{(x+1)^3 + 6e^x + 3x+5}\, dx }\]

If the value of the integral above can be expressed as \(\ln\left( \frac {Ae}{Be+C} \right) \) for positive integers \(A,B\) and \(C\), find the minimum value of \(A+B+C\).

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