Evaluate the following indefinite integral. \( \int x \sqrt{x^{3}} d x \) \( \int x \sqrt{x^{3}} d x=\square \)

Calculate the integral \( \int x \sqrt{x^{3}} dx \). Evaluate the integral by following steps: - step0: Evaluate using partial integration formula: \(\int x\sqrt{x^{3}} dx\) - step1: Simplify the root: \(\int x\times x\sqrt{x} dx\) - step2: Multiply the terms: \(\int x^{2}\sqrt{x} dx\) - step3: Prepare for integration by parts: \(\begin{align}&u=x^{2}\\&dv=\sqrt{x}dx\end{align}\) - step4: Calculate the derivative: \(\begin{align}&du=2xdx\\&dv=\sqrt{x}dx\end{align}\) - step5: Evaluate the integral: \(\begin{align}&du=2xdx\\&v=\frac{2}{3}x^{\frac{3}{2}}\end{align}\) - step6: Substitute the values into formula: \(x^{2}\times \frac{2}{3}x^{\frac{3}{2}}-\int 2x\times \frac{2}{3}x^{\frac{3}{2}} dx\) - step7: Calculate: \(\frac{2}{3}x^{\frac{7}{2}}-\int \frac{4}{3}x^{\frac{5}{2}} dx\) - step8: Evaluate the integral: \(\frac{2}{3}x^{\frac{7}{2}}-\frac{8}{21}x^{\frac{7}{2}}\) - step9: Collect like terms: \(\left(\frac{2}{3}-\frac{8}{21}\right)x^{\frac{7}{2}}\) - step10: Subtract the numbers: \(\frac{2}{7}x^{\frac{7}{2}}\) - step11: Simplify: \(\frac{2}{7}x^{3}\sqrt{x}\) - step12: Add the constant of integral C: \(\frac{2}{7}x^{3}\sqrt{x} + C, C \in \mathbb{R}\) The indefinite integral of \( \int x \sqrt{x^{3}} dx \) is \( \frac{2}{7}x^{3}\sqrt{x} + C \), where \( C \) is an arbitrary constant.