\begin{array}{l}
 \int {\frac{{dx}}{{Ax^2  + Bx + C}}}  = \frac{1}{A}\int {\frac{{dx}}{{x^2  + \frac{B}{A}x + \frac{C}{A}}}}  = \frac{1}{A}\int {\frac{{dx}}{{(x + \frac{B}{{2A}})^2  + \frac{C}{A} - \frac{{B^2 }}{{4A^2 }}}}}  = \frac{1}{A}\int {\frac{{dx}}{{(x + \frac{B}{{2A}})^2  + \frac{{4AC - B^2 }}{{4A^2 }}}}}  = \left| \begin{array}{l}
 x + \frac{B}{{2A}} = \sqrt {\frac{{4AC - B^2 }}{{4A^2 }}} t \\ 
 dx = \sqrt {\frac{{4AC - B^2 }}{{4A^2 }}} dt \\ 
 \end{array} \right| =  \\ 
  = \frac{1}{A}\int {\frac{{\sqrt {\frac{{4AC - B^2 }}{{4A^2 }}} dt}}{{\frac{{4AC - B^2 }}{{4A^2 }}t^2  + \frac{{4AC - B^2 }}{{4A^2 }}}}}  = \frac{1}{A}\frac{{\frac{{\sqrt {4AC - B^2 } }}{{2A}}}}{{\frac{{4AC - B^2 }}{{4A^2 }}}}\int {\frac{{dt}}{{t^2  + 1}}}  = \frac{2}{{\sqrt {4AC - B^2 } }}\int {\frac{{dt}}{{t^2  + 1}}}  = \frac{2}{{\sqrt {4AC - B^2 } }}{\mathop{\rm arctg}\nolimits} t = ... \\ 
 \end{array}