$\begin{aligned} & \int \frac{e^{3 \log 2 x}+5 e^{2 \log _e 2 x}}{e^{4 \log _e x}+5 e^{3 \log _e x}-7 e^{2 \log _e x}} d x, x>0 \\ & =\int \frac{(2 x)^3+5(2 x)^2}{x^4+5 x^3-7 x^2} d x=\int \frac{4 x^2(2 x+5)}{x^2\left(x^2+5 x-7\right)} d x \\ & =4 \int \frac{d\left(x^2+5 x-7\right)}{\left(x^2+5 x-7\right)}=4 \log _e\left|x^2+5 x-7\right|+c\end{aligned}$