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

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

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

                 b + 2 c x
  2 b ArcTan[-----------------]
                    2
             Sqrt[-b  + 4 a c]
-(----------------------------- - 2 Log[x] + 
               2
        Sqrt[-b  + 4 a c]
 
     Log[a + x (b + c x)]) / (2 a)
Time to compute: 0.20 second
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