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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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