Answer
$$\frac{{4{x^{7/4}}}}{7} + \frac{{2{x^{7/2}}}}{7} + C$$
Work Step by Step
$$\eqalign{
& \int {\left( {\root 4 \of {{x^3}} + \sqrt {{x^5}} } \right)} dx \cr
& {\text{use the radical property }}\root n \of {{x^m}} = {x^{m/n}} \cr
& = \int {\left( {{x^{3/4}} + {x^{5/2}}} \right)} dx \cr
& = \int {{x^{3/4}}} dx + \int {{x^{5/2}}} dx \cr
& {\text{find the antiderivative by the power rule}} \cr
& = \frac{{{x^{3/4 + 1}}}}{{3/4 + 1}} + \frac{{{x^{5/2 + 1}}}}{{5/2 + 1}} + C \cr
& {\text{simplify}} \cr
& = \frac{{{x^{7/4}}}}{{7/4}} + \frac{{{x^{7/2}}}}{{7/2}} + C \cr
& = \frac{{4{x^{7/4}}}}{7} + \frac{{2{x^{7/2}}}}{7} + C \cr} $$
