$\int {\sqrt x \left( {3{x^2} + 2x + 3} \right)dx}$

Let $I = \int {\sqrt x \left( {3{x^2} + 2x + 3} \right)dx} = \int {\left( {3{x^{5/2}} + 2{x^{3/2}} + 3{x^{1/2}}} \right)dx}$

$= \cfrac{{3{x^{7/2}}}}{{7/2}} + \cfrac{{2{x^{5/2}}}}{{5/2}} + \cfrac{{3{x^{3/2}}}}{{3/2}} + C = \cfrac{6}{7}{x^{7/2}} + \cfrac{4}{5}{x^{5/2}} + 2{x^{3/2}} + C$