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Integrate[x*Sqrt[a + b*x + c*x^2], x] ==
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 2*b*c*x + 8*c*(a + c*x^2)) + 3*(b^3 - 4*a*b*c)* Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/ (48*c^(5/2))



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

(2 Sqrt[c] Sqrt[a + x (b + c x)] 
 
          2                         2
     (-3 b  + 2 b c x + 8 c (a + c x )) + 
 
        3
    3 (b  - 4 a b c) Log[b + 2 c x + 
 
                                                5/2
       2 Sqrt[c] Sqrt[a + x (b + c x)]]) / (48 c   )

Time to compute: 0.27 second

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