Find the anti-derivative of LaTeX: \displaystyle f(x) = \frac{- 2 x^{3} + 4 x^{2} - 8 x - 5}{\sqrt[3]{x}}

Using termwise division gives LaTeX: \displaystyle f(x) = - 2 x^{\frac{8}{3}} + 4 x^{\frac{5}{3}} - 8 x^{\frac{2}{3}} - \frac{5}{\sqrt[3]{x}} . Finding the antiderivative of each term gives LaTeX: \displaystyle F(x) = - \frac{6 x^{\frac{11}{3}}}{11} + \frac{3 x^{\frac{8}{3}}}{2} - \frac{24 x^{\frac{5}{3}}}{5} - \frac{15 x^{\frac{2}{3}}}{2} + C