Answer
$$\frac{{{x^3}}}{3}{\left( {2x + 1} \right)^{3/2}} - \frac{1}{5}{x^2}{\left( {2x + 1} \right)^{5/2}} + \frac{2}{{32}}x{\left( {2x + 1} \right)^{7/2}} - \frac{2}{{315}}{\left( {2x + 1} \right)^{9/2}} + C$$
Work Step by Step
$$\eqalign{
& \int {{x^3}\sqrt {2x + 1} } dx \cr
& {\text{Evaluate using tabular integration by parts}} \cr
& \frac{d}{{dx}}{\text{ }}\int {dx} \cr
& {x^3}{\text{ + }}\sqrt {2x + 1} \cr
& 3{x^2}{\text{ }} - {\text{ }}\frac{1}{3}{\left( {2x + 1} \right)^{3/2}} \cr
& 6x{\text{ + }}\frac{1}{{15}}{\left( {2x + 1} \right)^{5/2}} \cr
& 6{\text{ }} - {\text{ }}\frac{1}{{105}}{\left( {2x + 1} \right)^{7/2}} \cr
& {\text{0 + }}\frac{1}{{945}}{\left( {2x + 1} \right)^{9/2}} \cr
& \cr
& {\text{Then,}} \cr
& = \frac{{{x^3}}}{3}{\left( {2x + 1} \right)^{3/2}} - \left( {\frac{1}{5}{x^2}} \right)\left( {{{\left( {2x + 1} \right)}^{5/2}}} \right) + 6x\left( {\frac{1}{{105}}{{\left( {2x + 1} \right)}^{7/2}}} \right) \cr
& - 6\left( {\frac{1}{{945}}{{\left( {2x + 1} \right)}^{9/2}}} \right) + C \cr
& {\text{Simplifying}} \cr
& = \frac{{{x^3}}}{3}{\left( {2x + 1} \right)^{3/2}} - \frac{1}{5}{x^2}{\left( {2x + 1} \right)^{5/2}} + \frac{2}{{32}}x{\left( {2x + 1} \right)^{7/2}} - \frac{2}{{315}}{\left( {2x + 1} \right)^{9/2}} + C \cr} $$
