\[ \large \int_1^e \dfrac{\sqrt[3]{x} }{ x (\sqrt x + \sqrt[3]{x} )} \, dx \]

If the value of the integral above is equal to \[ a- b \ln(c+ \sqrt[d]{e} ) + f \ln g, \]

where \(a,b,c,d,f\) and \(g\) are positive integers with \(b\) and \(g\) minimized, find \(a+b+c+d+f+g\).

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