Answer
$$\frac{{{x^2}}}{2} + \frac{{{x^5}}}{5} + C$$
Work Step by Step
$$\eqalign{
& \int {x\left( {1 + {x^3}} \right)dx} \cr
& {\text{multiply the integrand}} \cr
& = \int {\left( {x + {x^{3 + 1}}} \right)dx} \cr
& = \int {\left( {x + {x^4}} \right)dx} \cr
& {\text{power rule}} \cr
& = \frac{{{x^{1 + 1}}}}{{1 + 1}} + \frac{{{x^{4 + 1}}}}{{4 + 1}} + C \cr
& = \frac{{{x^2}}}{2} + \frac{{{x^5}}}{5} + C \cr} $$
