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Sqrt[1 + a*x^2]*((93*x)/128 + (163*a*x^3)/192 + (25*a^2*x^5)/48 + (a^3*x^7)/8) + (35*ArcSinh[Sqrt[a]*x])/(128*Sqrt[a])


Integrate[(1 + a*x^2)^(7/2), x] ==
Sqrt[1 + a*x^2]*((93*x)/128 + (163*a*x^3)/192 + (25*a^2*x^5)/48 + (a^3*x^7)/8) + (35*ArcSinh[Sqrt[a]*x])/(128*Sqrt[a])



                  2 7/2
Integrate[(1 + a x )   , x] == 

                              3       2  5    3  7
            2   93 x   163 a x    25 a  x    a  x
Sqrt[1 + a x ] (---- + -------- + -------- + -----) + 
                128      192         48        8
 
  35 ArcSinh[Sqrt[a] x]
  ---------------------
       128 Sqrt[a]

Time to compute: 0.09 second

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