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

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

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