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Integrate[x^2/Sqrt[a + b*x + c*x^2], x] ==
(2*Sqrt[c]*(-3*b + 2*c*x)*(a + x*(b + c*x)) + (3*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)]])/ (8*c^(5/2)*Sqrt[a + x*(b + c*x)])



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

(2 Sqrt[c] (-3 b + 2 c x) (a + x (b + c x)) + 
 
        2
    (3 b  - 4 a c) Sqrt[a + x (b + c x)] 
 
     Log[b + 2 c x + 2 Sqrt[c] Sqrt[a + x (b + c x)]])
 
          5/2
    / (8 c    Sqrt[a + x (b + c x)])

Time to compute: 0.23 second

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