What is int (2x^3-3x^2-4x-3 ) / (-5x^2+ 2 x -4 )dx?

Using division, we get

$\setminus \frac{2 {x}^{3} - 3 {x}^{2} - 4 x - 3}{- 5 {x}^{2} + 2 x - 4} = - \frac{2}{5} x + \setminus \frac{11 {x}^{2} + 28 x + 15}{5 \left(5 {x}^{2} - 2 x + 4\right)}$

$= - \frac{2}{5} x + \frac{11}{25} + \frac{81}{125} \setminus \frac{10 x - 2}{5 {x}^{2} - 2 x + 4} + \setminus \frac{317}{125} \setminus \frac{1}{5 {x}^{2} - 2 x + 4}$

$\setminus \therefore \setminus \int \setminus \frac{2 {x}^{3} - 3 {x}^{2} - 4 x - 3}{- 5 {x}^{2} + 2 x - 4} \setminus \mathrm{dx}$

$= \setminus \int \left(- \frac{2}{5} x + \frac{11}{25} + \frac{81}{125} \setminus \frac{10 x - 2}{5 {x}^{2} - 2 x + 4} + \setminus \frac{317}{125} \setminus \frac{1}{5 {x}^{2} - 2 x + 4}\right) \mathrm{dx}$

$= - \frac{2}{5} \setminus \int x \setminus \mathrm{dx} + \frac{11}{25} \setminus \int \mathrm{dx} + \frac{81}{125} \setminus \int \setminus \frac{\left(10 x - 2\right) \mathrm{dx}}{5 {x}^{2} - 2 x + 4} + \setminus \frac{317}{125} \setminus \int \setminus \frac{\mathrm{dx}}{5 {x}^{2} - 2 x + 4}$

$= - \frac{2}{5} \setminus \frac{{x}^{2}}{2} + \frac{11}{25} x + \frac{81}{125} \setminus \int \setminus \frac{d \left(5 {x}^{2} - 2 x + 4\right)}{5 {x}^{2} - 2 x + 4} + \setminus \frac{317}{625} \setminus \int \setminus \frac{\mathrm{dx}}{{\left(x - \frac{1}{5}\right)}^{2} + \frac{19}{25}}$

$= - \setminus \frac{{x}^{2}}{5} + \frac{11}{25} x + \frac{81}{125} \setminus \ln | 5 {x}^{2} - 2 x + 4 | + \setminus \frac{317}{625} \setminus \frac{1}{\setminus \frac{\sqrt{19}}{5}} \setminus {\tan}^{- 1} \left(\setminus \frac{x - \frac{1}{5}}{\setminus \frac{\sqrt{19}}{5}}\right) + C$

$= - \setminus \frac{{x}^{2}}{5} + \frac{11}{25} x + \frac{81}{125} \setminus \ln | 5 {x}^{2} - 2 x + 4 | + \setminus \frac{317}{125 \setminus \sqrt{19}} \setminus {\tan}^{- 1} \left(\setminus \frac{5 x - 1}{\setminus \sqrt{19}}\right) + C$