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


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



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

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

Time to compute: 0.17 second

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