${x^2}\log x$

Let $I = \int {{x^2}\log x\,dx}$
$= \log x\left( {\cfrac{{{x^3}}}{3}} \right) - \int {\left( {\left( {\cfrac{d}{{dx}}\left( {\log \,x} \right)} \right)\left( {\cfrac{{{x^3}}}{3}} \right)} \right)} dx$

$= \cfrac{{{x^3}}}{3}\log x - \cfrac{1}{3}\int {{x^2}dx} = \cfrac{{{x^3}}}{3}\log \left( x \right) - \cfrac{1}{3} \times \cfrac{{{x^3}}}{3} + C$

$= \cfrac{{{x^3}}}{3}\log x - \cfrac{{{x^3}}}{9} + C$