Single Variable Calculus, Chapter 5, 5.4, Section 5.4, Problem 6

Determine the general indefinite integral $\displaystyle \left( \sqrt{x^3} + \sqrt[3]{x^2} \right) dx$

$
\begin{equation}
\begin{aligned}
\int \left( \sqrt{x^3} + \sqrt[3]{x^2} \right) dx &= \int \left( x^{\frac{3}{2}} + x^{\frac{3}{2}} \right) dx\\
\\
\int \left( \sqrt{x^3} + \sqrt[3]{x^2} \right) dx &= \int x^{\frac{3}{2}} dx + \int x^{\frac{2}{3}} dx\\
\\
\int \left( \sqrt{x^3} + \sqrt[3]{x^2} \right) dx &= \frac{x^{\frac{3}{2}} + 1}{ \frac{3}{2} + 1 } + \frac{x^{\frac{2}{3}}+1}{\frac{2}{3}+1} + C\\
\\
\int \left( \sqrt{x^3} + \sqrt[3]{x^2} \right) dx &= \frac{x^{\frac{5}{2}}}{\frac{5}{2}} + \frac{x^{\frac{5}{3}}}{\frac{5}{3}} + C\\
\\
\int \left( \sqrt{x^3} + \sqrt[3]{x^2} \right) dx &= \frac{2x^{\frac{5}{2}}}{5} + \frac{3x^{\frac{5}{3}}}{5} + C \qquad \text{or} \qquad \int \left( \sqrt{x^3} + \sqrt[3]{x^2} \right) dx = \frac{2x^{\frac{5}{2}} + 3x^{\frac{5}{3}}}{5} + C
\end{aligned}
\end{equation}
$