$\int \frac{5 x^{3}+12}{x^{3}-5 x^{2}+4 x} d x$

Note that, $5 x^{3}+12 = 5* (x^{3}-5 x^{2}+4 x)+ 25x^2-20x-12$

$$\begin{eqnarray*} \int \frac{5 x^{3}+12}{x^{3}-5 x^{2}+4 x} d x &=&\int 5 \ dx+\int \frac{25x^2-20x-12}{x^{3}-5 x^{2}+4 x}\\ &=& 5x + \int \frac{A}{x}+ \int \frac{B}{x-4}+ \int \frac{C}{x-1}\\ &=& 5x -3\ln|x| + \ \frac{77}{3}\ln|x-4| \ + \frac{7}{3}\ln|x-1| +\ C \end{eqnarray*}$$