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(a*x*(3 - 2*a^2*x^2)*Sqrt[1 + a^2*x^2] + (-3 + 8*a^4*x^4)*ArcSinh[a*x])/(32*a^4)

Integrate[x^3*ArcSinh[a*x], x] ==
(a*x*(3 - 2*a^2*x^2)*Sqrt[1 + a^2*x^2] + (-3 + 8*a^4*x^4)*ArcSinh[a*x])/(32*a^4) 

           3
Integrate[x  ArcSinh[a x], x] == 

             2  2            2  2
(a x (3 - 2 a  x ) Sqrt[1 + a  x ] + 
 
             4  4                       4
    (-3 + 8 a  x ) ArcSinh[a x]) / (32 a )
Time to compute: 0.06 second
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