Find the anti-derivative of LaTeX: \displaystyle f(x) = \frac{- 6 x^{3} - 2 x^{2} + 5 x - 2}{\sqrt{x}}

Using termwise division gives LaTeX: \displaystyle f(x) = - 6 x^{\frac{5}{2}} - 2 x^{\frac{3}{2}} + 5 \sqrt{x} - \frac{2}{\sqrt{x}} . Finding the antiderivative of each term gives LaTeX: \displaystyle F(x) = - \frac{12 x^{\frac{7}{2}}}{7} - \frac{4 x^{\frac{5}{2}}}{5} + \frac{10 x^{\frac{3}{2}}}{3} - 4 \sqrt{x} + C