Answer
$$\frac{1}{5}\left( {x - 1} \right){\left( {3 + 2x} \right)^{3/2}} + C$$
Work Step by Step
$$\eqalign{
& \int {x\sqrt {2x + 3} } dx \cr
& {\text{Use the Endpaper Integral Table to evaluate the integral}} \cr
& {\text{Rewrite the integrand}} \cr
& = \int {x\sqrt {3 + 2x} } dx \cr
& {\text{The integrand has a expression in the form }}\sqrt {a + bu} {} \cr
& {\text{Use formula 102}} \cr
& \left( {102} \right):\,\,\,\,\int {u\sqrt {a + bu} du} = \frac{2}{{15{b^2}}}\left( {3bu - 2a} \right){\left( {a + bu} \right)^{3/2}} + C \cr
& {\text{let }}u = x,\,\,\,a = 3{\text{ and }}b = 2 \cr
& \int {x\sqrt {3 + 2x} } dx = \frac{2}{{15{{\left( 2 \right)}^2}}}\left( {3\left( 2 \right)x - 2\left( 3 \right)} \right){\left( {3 + 2x} \right)^{3/2}} + C \cr
& {\text{simplifying}} \cr
& \int {x\sqrt {2x + 3} } dx = \frac{1}{{30}}\left( {6x - 6} \right){\left( {3 + 2x} \right)^{3/2}} + C \cr
& \int {x\sqrt {2x + 3} } dx = \frac{6}{{30}}\left( {x - 1} \right){\left( {3 + 2x} \right)^{3/2}} + C \cr
& \int {x\sqrt {2x + 3} } dx = \frac{1}{5}\left( {x - 1} \right){\left( {3 + 2x} \right)^{3/2}} + C \cr} $$
