Find the anti-derivative of LaTeX: \displaystyle f(x) = \frac{5 x^{3} + 4 x^{2} + 7 x + 9}{\sqrt[3]{x}}

Using termwise division gives LaTeX: \displaystyle f(x) = 5 x^{\frac{8}{3}} + 4 x^{\frac{5}{3}} + 7 x^{\frac{2}{3}} + \frac{9}{\sqrt[3]{x}} . Finding the antiderivative of each term gives LaTeX: \displaystyle F(x) = \frac{15 x^{\frac{11}{3}}}{11} + \frac{3 x^{\frac{8}{3}}}{2} + \frac{21 x^{\frac{5}{3}}}{5} + \frac{27 x^{\frac{2}{3}}}{2} + C