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Integrate[x^2*Sqrt[a + b*x + c*x^2], x] ==
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]* (15*b^3 - 10*b^2*c*x + 24*c^2*x*(a + 2*c*x^2) + b*(-52*a*c + 8*c^2*x^2)) - 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)* Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/ (384*c^(7/2))



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

(2 Sqrt[c] Sqrt[a + x (b + c x)] 
 
          3       2           2             2
     (15 b  - 10 b  c x + 24 c  x (a + 2 c x ) + 
 
                       2  2
       b (-52 a c + 8 c  x )) - 
 
          4         2         2  2
    3 (5 b  - 24 a b  c + 16 a  c ) 
 
     Log[b + 2 c x + 2 Sqrt[c] Sqrt[a + x (b + c x)]])
 
            7/2
    / (384 c   )

Time to compute: 0.33 second

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