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Integrate[1/(x^2*(a + b*x + c*x^2)), x] ==
((-2*a)/x + (2*(b^2 - 2*a*c)* ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/ Sqrt[-b^2 + 4*a*c] - 2*b*Log[x] + b*Log[a + x*(b + c*x)])/(2*a^2)

                   1
Integrate[-------------------, x] == 
           2               2
          x  (a + b x + c x )

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