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-(((b + 2*c*x)/(a + x*(b + c*x)) + (4*c*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(b^2 - 4*a*c))

Integrate[(a + b*x + c*x^2)^(-2), x] ==
-(((b + 2*c*x)/(a + x*(b + c*x)) + (4*c*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/ Sqrt[-b^2 + 4*a*c])/(b^2 - 4*a*c)) 

                        2 -2
Integrate[(a + b x + c x )  , x] == 

                                   b + 2 c x
                    4 c ArcTan[-----------------]
                                      2
     b + 2 c x                 Sqrt[-b  + 4 a c]
  --------------- + -----------------------------
  a + x (b + c x)                2
                          Sqrt[-b  + 4 a c]
-(-----------------------------------------------)
                     2
                    b  - 4 a c
Time to compute: 0.22 second
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