$\int {\cfrac{{{x^3} + 3x + 4}}{{\sqrt x }}} dx$

: Let$I = \int {\left( {\cfrac{{{x^3} + 3x + 4}}{{\sqrt x }}} \right)} dx = \int {\left( {{x^{5/2}} + 3{x^{1/2}} + 4{x^{ - 1/2}}} \right)dx}$

$= \cfrac{{{x^{7/2}}}}{{7/2}} + \cfrac{{3{x^{3/2}}}}{{3/2}} + \cfrac{{4{x^{1/2}}}}{{1/2}} + C = \cfrac{2}{7}{x^{7/2}} + 2{x^{3/2}} + 8\sqrt x + C$