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Integrate[1/(x^3*(a + b*x)^(1/3)), x] ==
(-3*a^2 + a*b*x + 4*b^2*x^2 - 4*b^2*(1 + a/(b*x))^(1/3)*x^2*Hypergeometric2F1[ 1/3, 1/3, 4/3, -(a/(b*x))])/ (6*a^2*x^2*(a + b*x)^(1/3))



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

     2              2  2
(-3 a  + a b x + 4 b  x  - 
 
       2       a  1/3  2
    4 b  (1 + ---)    x  
              b x
 
                       1  1  4     a
     Hypergeometric2F1[-, -, -, -(---)]) / 
                       3  3  3    b x
 
      2  2          1/3
  (6 a  x  (a + b x)   )

Time to compute: 0.10 second

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