$\cfrac{{{e^{5\log x}} - {e^{4\log x}}}}{{{e^{3\log x}} - {e^{2\log x}}}}$

Let $I = \int {\cfrac{{{e^{5\log x}} - {e^{4\log x}}}}{{{e^{3\log x}} - {e^{2\log x}}}}dx} = \int {\left( {\cfrac{{{e^{\log {x^5}}} - {e^{\log {x^4}}}}}{{{e^{\log {x^3}}} - {e^{\log {x^2}}}}}} \right)dx}$

$= \int {\cfrac{{{x^5} - {x^4}}}{{{x^3} - {x^2}}}dx} = \int {\cfrac{{{x^4}\left( {x - 1} \right)}}{{{x^2}\left( {x - 1} \right)}}} dx = \int {{x^2}dx} = \cfrac{{{x^3}}}{3} + C$