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Integrate[x^2/(a + b*x + c*x^2)^(3/2), x] ==
(2*Sqrt[c]*(b^2*x + a*(b - 2*c*x)) - (b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)]* Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/ (c^(3/2)*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)])



                   2
                  x
Integrate[-------------------, x] == 
                        2 3/2
          (a + b x + c x )

             2
(2 Sqrt[c] (b  x + a (b - 2 c x)) - 
 
      2
    (b  - 4 a c) Sqrt[a + x (b + c x)] 
 
     Log[b + 2 c x + 2 Sqrt[c] Sqrt[a + x (b + c x)]])
 
        3/2    2
    / (c    (-b  + 4 a c) Sqrt[a + x (b + c x)])

Time to compute: 0.29 second

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