Integrate the function: $\frac{e^{5 \log x}-e^{4 \log x}}{e^{3 \log x}-e^{2 \log x}}$

Given the integral $I = \int \frac{e^{5 \log x}-e^{4 \log x}}{e^{3 \log x}-e^{2 \log x}} dx$.
Using the property $a \log b = \log b^a$ and $e^{\log x} = x$,we simplify the integrand:
$\frac{e^{5 \log x}-e^{4 \log x}}{e^{3 \log x}-e^{2 \log x}} = \frac{e^{\log x^5} - e^{\log x^4}}{e^{\log x^3} - e^{\log x^2}} = \frac{x^5 - x^4}{x^3 - x^2}$
Factor out the terms in the numerator and denominator:
$= \frac{x^4(x - 1)}{x^2(x - 1)}$
For $x \neq 1$ and $x \neq 0$,we can cancel $(x - 1)$ and $x^2$:
$= x^2$
Now,integrate the simplified function:
$\int x^2 dx = \frac{x^3}{3} + C$,where $C$ is the constant of integration.