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Integrate[1/(x^2*Sqrt[a + b*x + c*x^2]), x] ==
(-2*Sqrt[a]*(a + x*(b + c*x)) - b*x*Sqrt[a + x*(b + c*x)]*Log[Sqrt[a]*b*x] + b*x*Sqrt[a + x*(b + c*x)]* Log[-2*a*(2*a + b*x + 2*Sqrt[a]* Sqrt[a + x*(b + c*x)])])/ (2*a^(3/2)*x*Sqrt[a + x*(b + c*x)])



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

(-2 Sqrt[a] (a + x (b + c x)) - 
 
    b x Sqrt[a + x (b + c x)] Log[Sqrt[a] b x] + 
 
    b x Sqrt[a + x (b + c x)] 
 
     Log[-2 a (2 a + b x + 
 
         2 Sqrt[a] Sqrt[a + x (b + c x)])]) / 
 
      3/2
  (2 a    x Sqrt[a + x (b + c x)])

Time to compute: 0.27 second

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