Answer
$$F(x)= \frac{ 3}{5}x^{ 5/3}+\frac{2}{5}x^{ 5/2}+C$$
Work Step by Step
Given $$f(x) = \sqrt[3]{x^2}+x\sqrt{x}= x^{2/3}+x^{3/2}$$
Then by using if $f(x)=x^{\alpha}\ \to\ \ F(x) = \frac{x^{\alpha+1}}{\alpha+1}+C$
\begin{align*}
F(x) &= \frac{ 1}{5/3}x^{ 5/3}+\frac{1}{5/2}x^{ 5/2}+C\\
&= \frac{ 3}{5}x^{ 5/3}+\frac{2}{5}x^{ 5/2}+C
\end{align*}
To check
\begin{align*}
F'(x) &=x^{\frac{2}{3}}+x^{\frac{3}{2}}\\
&= \sqrt[3]{x^2}+x\sqrt{x}\\
&=f(x)
\end{align*}
