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


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



                           2
Integrate[Log[a + b x + c x ], x] == 

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

Time to compute: 0.09 second

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