Given that

\[\begin{cases} f(k,x)=\displaystyle \int \left(e^{x \ln k}+e^{k \ln x}+e^{k \ln k}\right) dx \\ f(e,0)=1 \\ f(4,3)=\dfrac{a}{b}+\dfrac{c}{\ln 2} \end{cases} \]

Find \(a-25 bc\), where \(a\), \(b\) and \(c\) are positive integers with \(a\) and \(b\) being coprime integers.

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