教科書基本レベル
$$\begin{align*}\int x\sqrt{2x+1}dx & =\int\frac{1}{2}\left(2x+1-1\right)\sqrt{2x+1}dx\\& =\frac{1}{2}\int\left\{ \left(2x+1\right)\sqrt{2x+1}-\sqrt{2x+1}\right\} dx\\& =\frac{1}{2}\left\{ \frac{2}{5}\left(2x+1\right)^{\frac{5}{2}}\cdot\frac{1}{2}-\frac{2}{3}\left(2x+1\right)^{\frac{3}{2}}\cdot\frac{1}{2}\right\} +C\\& =\frac{1}{2}\left(2x+1\right)^{\frac{3}{2}}\left\{ \frac{1}{5}\left(2x+1\right)-\frac{1}{3}\right\} +C\\& =\frac{1}{15}\left(3x-1\right)\left(2x+1\right)\sqrt{2x+1}+C\end{align*}$$
