$I=\int\limits_{1}^{e}\dfrac{\left(x^3+1\right)\ln x+2x^2+1}{2+x\ln x}dx=\int\limits_{1}^{e}\left ( x^2+\dfrac{1+\ln x}{2+x\ln x} \right )dx$
$=\int\limits_{1}^{e}\left ( x^2+\dfrac{d(2+x\ln x)}{2+x\ln x} \right )dx=\left[ { \frac13x^3+\ln|2+x\ln x| } \right]_{1}^{e}$