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Integrate[(a + b*x + c*x^2)^(3/2), x] ==
((b + 2*c*x)*Sqrt[a + x*(b + c*x)]* (-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^2)))/ (64*c^2) + (3*(b^2 - 4*a*c)^2* Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/ (128*c^(5/2))



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

((b + 2 c x) Sqrt[a + x (b + c x)] 
 
          2                             2           2
     (-3 b  + 8 b c x + 4 c (5 a + 2 c x ))) / (64 c )
 
           2         2
    + (3 (b  - 4 a c)  
 
     Log[b + 2 c x + 2 Sqrt[c] Sqrt[a + x (b + c x)]])
 
             5/2
     / (128 c   )

Time to compute: 0.16 second

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