The integral $\int \frac{e^{3{\log }_{e}2x}+5e^{2{\log }_{e}2x}}{e^{4{\log }_{e}x}+5e^{3{\log }_{e}x}-7e^{2{\log }_{e}x}}dx,x>0$ is equal to 

$\begin{array}{c}\int \frac{e^{3{\log }_{e}2x}+5e^{2{\log }_{e}2x}}{e^{4{\log }_{e}x}+5e^{3{\log }_{e}x}-7e^{2{\log }_{e}x}}dx=\int \frac{8x^3+5·4x^2}{x^4+5x^3-7x^2}dx\\ \\ =\int \frac{8x+20}{x^2+5x-7}dx\\ =\boxed{4\log \left|x^2+5x-7\right|+C}\end{array}\left( \sin ce {\int }_{ }^{ }\frac{f^{\text{'}}\left(x\right)}{f\left(x\right)}dx=\log \left|f\left(x\right)\right|+c\right)$