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

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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