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Calculus Level 5

Suppose that there are two functions \( f(x) \) and \( g(x) \), such that the derivative of the product of the two functions is

\[ \left(5x^2 + \frac{107}{6} x + \dfrac {47}{3} \right)e^{5x} \]

and that their Wronskian is

\[ \left(-x^2 - \frac{19}{6}x - \frac {7}{3}\right)e^{5x} \]

Suppose also that \( \dfrac { f'(x)}{g'(x)} = \dfrac{3}{2}e^x \). Assuming all arbitrary constants of integration to be zero, if the value of \( (f+g)(1) \) can be expressed in the form \[\dfrac {Ae^B}{C} + \dfrac {De^F}{G} \] where \(A,B,C,D,F\) and \(G\) are positive integers with \( \gcd(A,C) = \gcd(D,G) = 1\), determine the value of \( A + B + C + D + F + G \).

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