Answer
$$\frac{{25{s^3}}}{3} + 15{s^2} + 9s + C$$
Work Step by Step
$$\eqalign{
& \int {{{\left( {5s + 3} \right)}^2}} ds \cr
& {\text{expanding the integrand}} \cr
& = \int {\left( {25{s^2} + 30s + 9} \right)} ds \cr
& {\text{by the power rule for indefinite integrals}} \cr
& = \frac{{25{s^{2 + 1}}}}{{2 + 1}} + \frac{{30{s^{1 + 1}}}}{{1 + 1}} + 9s + C \cr
& = \frac{{25{s^3}}}{3} + \frac{{30{s^2}}}{2} + 9s + C \cr
& {\text{simplify}} \cr
& = \frac{{25{s^3}}}{3} + 15{s^2} + 9s + C \cr
& \cr
& \cr
& {\text{check the antiderivative by differentiation}} \cr
& {\text{ = }}\frac{d}{{ds}}\left( {\frac{{25{s^3}}}{3} + 15{s^2} + 9s + C} \right) \cr
& {\text{ = }}\frac{{25\left( 3 \right){s^2}}}{3} + 15\left( 2 \right)s + 9\left( 1 \right) + 0 \cr
& {\text{ = }}25{s^2} + 30s + 9 \cr
& {\text{factor}} \cr
& = {\left( {5s + 3} \right)^2} \cr} $$
