Mathematics - Indefinite Integration Question with Solution | TestHub

$\int \frac{{\mathrm{e}}^{3{\log }_{e}2x}+5{\mathrm{e}}^{2{\log }_{e}2x}}{{\mathrm{e}}^{4{\log }_{e}x}+5{\mathrm{e}}^{3{\log }_{e}x}-7{\mathrm{e}}^{2{\log }_{e}x}}dx,x>0$

$=\int \frac{{\left(2x\right)}^3+5{\left(2x\right)}^2}{x^4+5x^3-7x^2}dx=\int \frac{4x^2\left(2x+5\right)}{x^2\left(x^2+5x-7\right)}dx$

$=4\int \frac{d\left(x^2+5x-7\right)}{\left(x^2+5x-7\right)}=4{\log }_e\left|x^2+5x-7\right|+c$