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Verified answer

Делаем замену:

x=t^6 (t=x^(1/6) )

dx=6t^5*dt

int( (t^6+t^4+t) * 6t^5 * dt/ t^6 *(1+t^2) )=6*int( (t^5+t^3+1)*dt/(1+t^2) )=

=6*( int( t^3 *dt) +int(dt/(1+t^2) )=6*t^4/4  +6*arctg(t) +c=

3/2  *  x^(2/3) +6*arctg(x^(1/6) )+c