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Integrate[x*(a + b*x + c*x^2)^(3/2), x] ==
(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]* (15*b^4 - 10*b^3*c*x + 128*c^2*(a + c*x^2)^2 + 4*b^2*c*(-25*a + 2*c*x^2) + 8*b*c^2*x*(7*a + 22*c*x^2)) - 15*b*(b^2 - 4*a*c)^2* Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/ (1280*c^(7/2))



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

(2 Sqrt[c] Sqrt[a + x (b + c x)] 
 
          4       3            2         2 2
     (15 b  - 10 b  c x + 128 c  (a + c x )  + 
 
          2                 2
       4 b  c (-25 a + 2 c x ) + 
 
            2                2
       8 b c  x (7 a + 22 c x )) - 
 
           2         2
    15 b (b  - 4 a c)  
 
     Log[b + 2 c x + 2 Sqrt[c] Sqrt[a + x (b + c x)]])
 
             7/2
    / (1280 c   )

Time to compute: 0.44 second

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