Find the anti-derivative of LaTeX: \displaystyle f(x) = \frac{2 x^{3} - 4 x^{2} - 8 x - 9}{\sqrt[4]{x}}

Using termwise division gives LaTeX: \displaystyle f(x) = 2 x^{\frac{11}{4}} - 4 x^{\frac{7}{4}} - 8 x^{\frac{3}{4}} - \frac{9}{\sqrt[4]{x}} . Finding the antiderivative of each term gives LaTeX: \displaystyle F(x) = \frac{8 x^{\frac{15}{4}}}{15} - \frac{16 x^{\frac{11}{4}}}{11} - \frac{32 x^{\frac{7}{4}}}{7} - 12 x^{\frac{3}{4}} + C