The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, see lists of integrals. Throughout this article the constant of integration is omitted for brevity.
Integrals involving s = √x2 − a2
Assume x2 > a2 (for x2 < a2, see next section):



Here
where the positive value of
is to be taken.













![{\displaystyle \int {\frac {dx}{s^{5))}={\frac {1}{a^{4))}\left[{\frac {x}{s))-{\frac {1}{3)){\frac {x^{3)){s^{3))}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/054a5959ce5e03cf279c1b29dff2ba014ac6dcde)
![\int {\frac {dx}{s^{7))}=-{\frac {1}{a^{6))}\left[{\frac {x}{s))-{\frac {2}{3)){\frac {x^{3)){s^{3))}+{\frac {1}{5)){\frac {x^{5)){s^{5))}\right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/86843311de7fc72bc01f87742445f7c4b88899e9)
![\int {\frac {dx}{s^{9))}={\frac {1}{a^{8))}\left[{\frac {x}{s))-{\frac {3}{3)){\frac {x^{3)){s^{3))}+{\frac {3}{5)){\frac {x^{5)){s^{5))}-{\frac {1}{7)){\frac {x^{7)){s^{7))}\right]](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca32b3a8d7f9040840f5d1de3467129edff0d80b)

![{\displaystyle \int {\frac {x^{2}\,dx}{s^{7))}={\frac {1}{a^{4))}\left[{\frac {1}{3)){\frac {x^{3)){s^{3))}-{\frac {1}{5)){\frac {x^{5)){s^{5))}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a98057cf3f3d6b7025114445c972bb6b7b7af9d)
![{\displaystyle \int {\frac {x^{2}\,dx}{s^{9))}=-{\frac {1}{a^{6))}\left[{\frac {1}{3)){\frac {x^{3)){s^{3))}-{\frac {2}{5)){\frac {x^{5)){s^{5))}+{\frac {1}{7)){\frac {x^{7)){s^{7))}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cce4b87e7a47ce42042803038139f830afd5d37)
Integrals involving R = √ax2 + bx + c
Assume (ax2 + bx + c) cannot be reduced to the following expression (px + q)2 for some p and q.





















