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


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



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

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

Time to compute: 0.10 second

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