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

Using termwise division gives LaTeX: \displaystyle f(x) = - 5 x^{\frac{11}{4}} + 5 x^{\frac{7}{4}} - 2 x^{\frac{3}{4}} + \frac{8}{\sqrt[4]{x}} . Finding the antiderivative of each term gives LaTeX: \displaystyle F(x) = - \frac{4 x^{\frac{15}{4}}}{3} + \frac{20 x^{\frac{11}{4}}}{11} - \frac{8 x^{\frac{7}{4}}}{7} + \frac{32 x^{\frac{3}{4}}}{3} + C