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


Integrate[1/(x^3*(a^2 + x^2)^(3/2)), x] ==
-((a^3 + 3*a*x^2)/(x^2*Sqrt[a^2 + x^2]) + 3*Log[3*a*x] - 3*Log[-4*a^5*(a + Sqrt[a^2 + x^2])])/(2*a^5)



                 1
Integrate[---------------, x] == 
           3   2    2 3/2
          x  (a  + x )

     3        2
    a  + 3 a x
-(---------------- + 3 Log[3 a x] - 
   2       2    2
  x  Sqrt[a  + x ]
 
               5            2    2           5
     3 Log[-4 a  (a + Sqrt[a  + x ])]) / (2 a )

Time to compute: 0.24 second

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