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(x^(1 + n)*(-2 + 2*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, x^2/a^2] + (1 + n)*Log[-a^2 + x^2]))/(1 + n)^2


Integrate[x^n*Log[-a^2 + x^2], x] ==
(x^(1 + n)*(-2 + 2*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, x^2/a^2] + (1 + n)*Log[-a^2 + x^2]))/ (1 + n)^2



           n       2    2
Integrate[x  Log[-a  + x ], x] == 

  1 + n                              1 + n  3 + n
(x      (-2 + 2 Hypergeometric2F1[1, -----, -----, 
                                       2      2
 
         2
        x                   2    2             2
        --] + (1 + n) Log[-a  + x ])) / (1 + n)
         2
        a

Time to compute: 0.08 second

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