Answer
$$\root 3 \of {{x^2}} + \sqrt {{x^3}} + C$$
Work Step by Step
$$\eqalign{
& \int {\left( {\root 3 \of {{x^2}} + \sqrt {{x^3}} } \right)dx} \cr
& {\text{rewritting radicals}} \cr
& = \int {\left( {{x^{2/3}} + {x^{3/2}}} \right)dx} \cr
& {\text{sum rule}} \cr
& = \int {{x^{2/3}}dx} + \int {{x^{3/2}}dx} \cr
& {\text{by the power rule}} \cr
& = \frac{{{x^{5/3}}}}{{5/3}} + \frac{{{x^{5/2}}}}{{5/2}} + C \cr
& simplify \cr
& = \frac{3}{5}{x^{5/3}} + \frac{2}{5}{x^{5/2}} + C \cr
& {\text{check by differentiation}} \cr
& {\text{ = }}\frac{d}{{dx}}\left( {\frac{3}{5}{x^{5/3}} + \frac{2}{5}{x^{5/2}} + C} \right) \cr
& {\text{ = }}\frac{3}{5}\left( {\frac{5}{3}} \right){x^{2/3}} + \frac{2}{5}\left( {\frac{5}{2}} \right){x^{3/2}} + C \cr
& {\text{simplify}} \cr
& {\text{ = }}{x^{2/3}} + {x^{3/2}} + C \cr
& = \root 3 \of {{x^2}} + \sqrt {{x^3}} + C \cr} $$
