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



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

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

Time to compute: 0.17 second

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