List of integrals of rational functions

Miscellaneous integrands

∫ f ′ ( x ) f ( x ) d x = ln ⁡ | f ( x ) | + C ∫ 1 x 2 + a 2 d x = 1 a arctan ⁡ x a + C ∫ 1 x 2 − a 2 d x = { − 1 a arctanh ⁡ x a = 1 2 a ln ⁡ a − x a + x + C (for  | x | < | a | ) − 1 a arccoth ⁡ x a = 1 2 a ln ⁡ x − a x + a + C (for  | x | > | a | ) ∫ d x x 2 n + 1 = ∑ k = 1 2 n − 1 { 1 2 n − 1 [ sin ⁡ ( ( 2 k − 1 ) π 2 n ) arctan ⁡ [ ( x − cos ⁡ ( ( 2 k − 1 ) π 2 n ) ) csc ⁡ ( ( 2 k − 1 ) π 2 n ) ] ] − 1 2 n [ cos ⁡ ( ( 2 k − 1 ) π 2 n ) ln ⁡ | x 2 − 2 x cos ⁡ ( ( 2 k − 1 ) π 2 n ) + 1 | ] } + C

Any rational function can be integrated using partial fractions in integration, by decomposing the rational function into a sum of functions of the form:

a ( x − b ) n , and a x + b ( ( x − c ) 2 + d 2 ) n .

Integrands of the form xm(a x + b)n

∫ 1 a x + b d x = 1 a ln ⁡ | a x + b | + C More generally, ∫ 1 a x + b d x = { 1 a ln ⁡ | a x + b | + C − a x + b < 0 1 a ln ⁡ | a x + b | + C + a x + b > 0 ∫ ( a x + b ) n d x = ( a x + b ) n + 1 a ( n + 1 ) + C (for  n ≠ − 1 ) (Cavalieri's quadrature formula) ∫ x a x + b d x = x a − b a 2 ln ⁡ | a x + b | + C ∫ x ( a x + b ) 2 d x = b a 2 ( a x + b ) + 1 a 2 ln ⁡ | a x + b | + C ∫ x ( a x + b ) n d x = a ( 1 − n ) x − b a 2 ( n − 1 ) ( n − 2 ) ( a x + b ) n − 1 + C (for  n ∉ { 1 , 2 } ) ∫ x ( a x + b ) n d x = a ( n + 1 ) x − b a 2 ( n + 1 ) ( n + 2 ) ( a x + b ) n + 1 + C (for  n ∉ { − 1 , − 2 } ) ∫ x 2 a x + b d x = b 2 ln ⁡ ( | a x + b | ) a 3 + a x 2 − 2 b x 2 a 2 + C ∫ x 2 ( a x + b ) 2 d x = 1 a 3 ( a x − 2 b ln ⁡ | a x + b | − b 2 a x + b ) + C ∫ x 2 ( a x + b ) 3 d x = 1 a 3 ( ln ⁡ | a x + b | + 2 b a x + b − b 2 2 ( a x + b ) 2 ) + C ∫ x 2 ( a x + b ) n d x = 1 a 3 ( − ( a x + b ) 3 − n ( n − 3 ) + 2 b ( a x + b ) 2 − n ( n − 2 ) − b 2 ( a x + b ) 1 − n ( n − 1 ) ) + C (for  n ∉ { 1 , 2 , 3 } ) ∫ 1 x ( a x + b ) d x = − 1 b ln ⁡ | a x + b x | + C ∫ 1 x 2 ( a x + b ) d x = − 1 b x + a b 2 ln ⁡ | a x + b x | + C ∫ 1 x 2 ( a x + b ) 2 d x = − a ( 1 b 2 ( a x + b ) + 1 a b 2 x − 2 b 3 ln ⁡ | a x + b x | ) + C

Integrands of the form xm / (a x2 + b x + c)n

For a ≠ 0 :

∫ 1 a x 2 + b x + c d x = { 2 4 a c − b 2 arctan ⁡ 2 a x + b 4 a c − b 2 + C (for  4 a c − b 2 > 0 ) 1 b 2 − 4 a c ln ⁡ | 2 a x + b − b 2 − 4 a c 2 a x + b + b 2 − 4 a c | + C = { − 2 b 2 − 4 a c a r c t a n h 2 a x + b b 2 − 4 a c + C (for  | 2 a x + b | < b 2 − 4 a c ) − 2 b 2 − 4 a c a r c c o t h 2 a x + b b 2 − 4 a c + C (else) (for  4 a c − b 2 < 0 ) − 2 2 a x + b + C (for  4 a c − b 2 = 0 ) ∫ x a x 2 + b x + c d x = 1 2 a ln ⁡ | a x 2 + b x + c | − b 2 a ∫ d x a x 2 + b x + c + C ∫ m x + n a x 2 + b x + c d x = { m 2 a ln ⁡ | a x 2 + b x + c | + 2 a n − b m a 4 a c − b 2 arctan ⁡ 2 a x + b 4 a c − b 2 + C (for  4 a c − b 2 > 0 ) m 2 a ln ⁡ | a x 2 + b x + c | − 2 a n − b m a b 2 − 4 a c a r c t a n h 2 a x + b b 2 − 4 a c + C (for  4 a c − b 2 < 0 ) m 2 a ln ⁡ | a x 2 + b x + c | − 2 a n − b m a ( 2 a x + b ) + C (for  4 a c − b 2 = 0 ) ∫ 1 ( a x 2 + b x + c ) n d x = 2 a x + b ( n − 1 ) ( 4 a c − b 2 ) ( a x 2 + b x + c ) n − 1 + ( 2 n − 3 ) 2 a ( n − 1 ) ( 4 a c − b 2 ) ∫ 1 ( a x 2 + b x + c ) n − 1 d x + C ∫ x ( a x 2 + b x + c ) n d x = − b x + 2 c ( n − 1 ) ( 4 a c − b 2 ) ( a x 2 + b x + c ) n − 1 − b ( 2 n − 3 ) ( n − 1 ) ( 4 a c − b 2 ) ∫ 1 ( a x 2 + b x + c ) n − 1 d x + C ∫ 1 x ( a x 2 + b x + c ) d x = 1 2 c ln ⁡ | x 2 a x 2 + b x + c | − b 2 c ∫ 1 a x 2 + b x + c d x + C

Integrands of the form xm (a + b xn)p

The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.

These reduction formulas can be used for integrands having integer and/or fractional exponents.

∫ x m ( a + b x n ) p d x = x m + 1 ( a + b x n ) p m + n p + 1 + a n p m + n p + 1 ∫ x m ( a + b x n ) p − 1 d x ∫ x m ( a + b x n ) p d x = − x m + 1 ( a + b x n ) p + 1 a n ( p + 1 ) + m + n ( p + 1 ) + 1 a n ( p + 1 ) ∫ x m ( a + b x n ) p + 1 d x ∫ x m ( a + b x n ) p d x = x m + 1 ( a + b x n ) p m + 1 − b n p m + 1 ∫ x m + n ( a + b x n ) p − 1 d x ∫ x m ( a + b x n ) p d x = x m − n + 1 ( a + b x n ) p + 1 b n ( p + 1 ) − m − n + 1 b n ( p + 1 ) ∫ x m − n ( a + b x n ) p + 1 d x ∫ x m ( a + b x n ) p d x = x m − n + 1 ( a + b x n ) p + 1 b ( m + n p + 1 ) − a ( m − n + 1 ) b ( m + n p + 1 ) ∫ x m − n ( a + b x n ) p d x ∫ x m ( a + b x n ) p d x = x m + 1 ( a + b x n ) p + 1 a ( m + 1 ) − b ( m + n ( p + 1 ) + 1 ) a ( m + 1 ) ∫ x m + n ( a + b x n ) p d x

Integrands of the form (A + B x) (a + b x)m (c + d x)n (e + f x)p

The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m, n and p toward 0.

These reduction formulas can be used for integrands having integer and/or fractional exponents.

Special cases of these reductions formulas can be used for integrands of the form ( a + b x ) m ( c + d x ) n ( e + f x ) p by setting B to 0.

∫ ( A + B x ) ( a + b x ) m ( c + d x ) n ( e + f x ) p d x = − ( A b − a B ) ( a + b x ) m + 1 ( c + d x ) n ( e + f x ) p + 1 b ( m + 1 ) ( a f − b e ) + 1 b ( m + 1 ) ( a f − b e ) ⋅ ∫ ( A + B x ) ( a + b x ) m ( c + d x ) n ( e + f x ) p d x = B ( a + b x ) m ( c + d x ) n + 1 ( e + f x ) p + 1 d f ( m + n + p + 2 ) + 1 d f ( m + n + p + 2 ) ⋅ ∫ ( A + B x ) ( a + b x ) m ( c + d x ) n ( e + f x ) p d x = ( A b − a B ) ( a + b x ) m + 1 ( c + d x ) n + 1 ( e + f x ) p + 1 ( m + 1 ) ( a d − b c ) ( a f − b e ) + 1 ( m + 1 ) ( a d − b c ) ( a f − b e ) ⋅

Integrands of the form xm (A + B xn) (a + b xn)p (c + d xn)q

The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m, p and q toward 0.

These reduction formulas can be used for integrands having integer and/or fractional exponents.

Special cases of these reductions formulas can be used for integrands of the form ( a + b x n ) p ( c + d x n ) q and x m ( a + b x n ) p ( c + d x n ) q by setting m and/or B to 0.

∫ x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = − ( A b − a B ) x m + 1 ( a + b x n ) p + 1 ( c + d x n ) q a b n ( p + 1 ) + 1 a b n ( p + 1 ) ⋅ ∫ x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = B x m + 1 ( a + b x n ) p + 1 ( c + d x n ) q b ( m + n ( p + q + 1 ) + 1 ) + 1 b ( m + n ( p + q + 1 ) + 1 ) ⋅ ∫ x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = − ( A b − a B ) x m + 1 ( a + b x n ) p + 1 ( c + d x n ) q + 1 a n ( b c − a d ) ( p + 1 ) + 1 a n ( b c − a d ) ( p + 1 ) ⋅ ∫ x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = B x m − n + 1 ( a + b x n ) p + 1 ( c + d x n ) q + 1 b d ( m + n ( p + q + 1 ) + 1 ) − 1 b d ( m + n ( p + q + 1 ) + 1 ) ⋅ ∫ x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = A x m + 1 ( a + b x n ) p + 1 ( c + d x n ) q + 1 a c ( m + 1 ) + 1 a c ( m + 1 ) ⋅ ∫ x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = A x m + 1 ( a + b x n ) p + 1 ( c + d x n ) q a ( m + 1 ) − 1 a ( m + 1 ) ⋅ ∫ x m ( A + B x n ) ( a + b x n ) p ( c + d x n ) q d x = ( A b − a B ) x m − n + 1 ( a + b x n ) p + 1 ( c + d x n ) q + 1 b n ( b c − a d ) ( p + 1 ) − 1 b n ( b c − a d ) ( p + 1 ) ⋅

Integrands of the form (d + e x)m (a + b x + c x2)p when b2 − 4 a c = 0

The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.

These reduction formulas can be used for integrands having integer and/or fractional exponents.

Special cases of these reductions formulas can be used for integrands of the form ( a + b x + c x 2 ) p when b 2 − 4 a c = 0 by setting m to 0.

∫ ( d + e x ) m ( a + b x + c x 2 ) p d x = ( d + e x ) m + 1 ( a + b x + c x 2 ) p e ( m + 1 ) − p ( d + e x ) m + 2 ( b + 2 c x ) ( a + b x + c x 2 ) p − 1 e 2 ( m + 1 ) ( m + 2 p + 1 ) + p ( 2 p − 1 ) ( 2 c d − b e ) e 2 ( m + 1 ) ( m + 2 p + 1 ) ∫ ( d + e x ) m + 1 ( a + b x + c x 2 ) p − 1 d x ∫ ( d + e x ) m ( a + b x + c x 2 ) p d x = ( d + e x ) m + 1 ( a + b x + c x 2 ) p e ( m + 1 ) − p ( d + e x ) m + 2 ( b + 2 c x ) ( a + b x + c x 2 ) p − 1 e 2 ( m + 1 ) ( m + 2 ) + 2 c p ( 2 p − 1 ) e 2 ( m + 1 ) ( m + 2 ) ∫ ( d + e x ) m + 2 ( a + b x + c x 2 ) p − 1 d x ∫ ( d + e x ) m ( a + b x + c x 2 ) p d x = − e ( m + 2 p + 2 ) ( d + e x ) m ( a + b x + c x 2 ) p + 1 ( p + 1 ) ( 2 p + 1 ) ( 2 c d − b e ) + ( d + e x ) m + 1 ( b + 2 c x ) ( a + b x + c x 2 ) p ( 2 p + 1 ) ( 2 c d − b e ) + e 2 m ( m + 2 p + 2 ) ( p + 1 ) ( 2 p + 1 ) ( 2 c d − b e ) ∫ ( d + e x ) m − 1 ( a + b x + c x 2 ) p + 1 d x ∫ ( d + e x ) m ( a + b x + c x 2 ) p d x = − e m ( d + e x ) m − 1 ( a + b x + c x 2 ) p + 1 2 c ( p + 1 ) ( 2 p + 1 ) + ( d + e x ) m ( b + 2 c x ) ( a + b x + c x 2 ) p 2 c ( 2 p + 1 ) + e 2 m ( m − 1 ) 2 c ( p + 1 ) ( 2 p + 1 ) ∫ ( d + e x ) m − 2 ( a + b x + c x 2 ) p + 1 d x ∫ ( d + e x ) m ( a + b x + c x 2 ) p d x = ( d + e x ) m + 1 ( a + b x + c x 2 ) p e ( m + 2 p + 1 ) − p ( 2 c d − b e ) ( d + e x ) m + 1 ( b + 2 c x ) ( a + b x + c x 2 ) p − 1 2 c e 2 ( m + 2 p ) ( m + 2 p + 1 ) + p ( 2 p − 1 ) ( 2 c d − b e ) 2 2 c e 2 ( m + 2 p ) ( m + 2 p + 1 ) ∫ ( d + e x ) m ( a + b x + c x 2 ) p − 1 d x ∫ ( d + e x ) m ( a + b x + c x 2 ) p d x = − 2 c e ( m + 2 p + 2 ) ( d + e x ) m + 1 ( a + b x + c x 2 ) p + 1 ( p + 1 ) ( 2 p + 1 ) ( 2 c d − b e ) 2 + ( d + e x ) m + 1 ( b + 2 c x ) ( a + b x + c x 2 ) p ( 2 p + 1 ) ( 2 c d − b e ) + 2 c e 2 ( m + 2 p + 2 ) ( m + 2 p + 3 ) ( p + 1 ) ( 2 p + 1 ) ( 2 c d − b e ) 2 ∫ ( d + e x ) m ( a + b x + c x 2 ) p + 1 d x ∫ ( d + e x ) m ( a + b x + c x 2 ) p d x = ( d + e x ) m ( b + 2 c x ) ( a + b x + c x 2 ) p 2 c ( m + 2 p + 1 ) + m ( 2 c d − b e ) 2 c ( m + 2 p + 1 ) ∫ ( d + e x ) m − 1 ( a + b x + c x 2 ) p d x ∫ ( d + e x ) m ( a + b x + c x 2 ) p d x = − ( d + e x ) m + 1 ( b + 2 c x ) ( a + b x + c x 2 ) p ( m + 1 ) ( 2 c d − b e ) + 2 c ( m + 2 p + 2 ) ( m + 1 ) ( 2 c d − b e ) ∫ ( d + e x ) m + 1 ( a + b x + c x 2 ) p d x

Integrands of the form (d + e x)m (A + B x) (a + b x + c x2)p

The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.

These reduction formulas can be used for integrands having integer and/or fractional exponents.

Special cases of these reductions formulas can be used for integrands of the form ( a + b x + c x 2 ) p and ( d + e x ) m ( a + b x + c x 2 ) p by setting m and/or B to 0.

∫ ( d + e x ) m ( A + B x ) ( a + b x + c x 2 ) p d x = ( d + e x ) m + 1 ( A e ( m + 2 p + 2 ) − B d ( 2 p + 1 ) + e B ( m + 1 ) x ) ( a + b x + c x 2 ) p e 2 ( m + 1 ) ( m + 2 p + 2 ) + 1 e 2 ( m + 1 ) ( m + 2 p + 2 ) p ⋅ ∫ ( d + e x ) m ( A + B x ) ( a + b x + c x 2 ) p d x = ( d + e x ) m ( A b − 2 a B − ( b B − 2 A c ) x ) ( a + b x + c x 2 ) p + 1 ( p + 1 ) ( b 2 − 4 a c ) + 1 ( p + 1 ) ( b 2 − 4 a c ) ⋅ ∫ ( d + e x ) m ( A + B x ) ( a + b x + c x 2 ) p d x = ( d + e x ) m + 1 ( A c e ( m + 2 p + 2 ) − B ( c d + 2 c d p − b e p ) + B c e ( m + 2 p + 1 ) x ) ( a + b x + c x 2 ) p c e 2 ( m + 2 p + 1 ) ( m + 2 p + 2 ) − p c e 2 ( m + 2 p + 1 ) ( m + 2 p + 2 ) ⋅ ∫ ( d + e x ) m ( A + B x ) ( a + b x + c x 2 ) p d x = ( d + e x ) m + 1 ( A ( b c d − b 2 e + 2 a c e ) − a B ( 2 c d − b e ) + c ( A ( 2 c d − b e ) − B ( b d − 2 a e ) ) x ) ( a + b x + c x 2 ) p + 1 ( p + 1 ) ( b 2 − 4 a c ) ( c d 2 − b d e + a e 2 ) + ∫ ( d + e x ) m ( A + B x ) ( a + b x + c x 2 ) p d x = B ( d + e x ) m ( a + b x + c x 2 ) p + 1 c ( m + 2 p + 2 ) + 1 c ( m + 2 p + 2 ) ⋅ ∫ ( d + e x ) m ( A + B x ) ( a + b x + c x 2 ) p d x = − ( B d − A e ) ( d + e x ) m + 1 ( a + b x + c x 2 ) p + 1 ( m + 1 ) ( c d 2 − b d e + a e 2 ) + 1 ( m + 1 ) ( c d 2 − b d e + a e 2 ) ⋅

Integrands of the form xm (a + b xn + c x2n)p when b2 − 4 a c = 0

The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.

These reduction formulas can be used for integrands having integer and/or fractional exponents.

Special cases of these reductions formulas can be used for integrands of the form ( a + b x n + c x 2 n ) p when b 2 − 4 a c = 0 by setting m to 0.

∫ x m ( a + b x n + c x 2 n ) p d x = x m + 1 ( a + b x n + c x 2 n ) p m + 2 n p + 1 + n p x m + 1 ( 2 a + b x n ) ( a + b x n + c x 2 n ) p − 1 ( m + 1 ) ( m + 2 n p + 1 ) − b n 2 p ( 2 p − 1 ) ( m + 1 ) ( m + 2 n p + 1 ) ∫ x m + n ( a + b x n + c x 2 n ) p − 1 d x ∫ x m ( a + b x n + c x 2 n ) p d x = ( m + n ( 2 p − 1 ) + 1 ) x m + 1 ( a + b x n + c x 2 n ) p ( m + 1 ) ( m + n + 1 ) + n p x m + 1 ( 2 a + b x n ) ( a + b x n + c x 2 n ) p − 1 ( m + 1 ) ( m + n + 1 ) + 2 c p n 2 ( 2 p − 1 ) ( m + 1 ) ( m + n + 1 ) ∫ x m + 2 n ( a + b x n + c x 2 n ) p − 1 d x ∫ x m ( a + b x n + c x 2 n ) p d x = ( m + n ( 2 p + 1 ) + 1 ) x m − n + 1 ( a + b x n + c x 2 n ) p + 1 b n 2 ( p + 1 ) ( 2 p + 1 ) − x m + 1 ( b + 2 c x n ) ( a + b x n + c x 2 n ) p b n ( 2 p + 1 ) − ( m − n + 1 ) ( m + n ( 2 p + 1 ) + 1 ) b n 2 ( p + 1 ) ( 2 p + 1 ) ∫ x m − n ( a + b x n + c x 2 n ) p + 1 d x ∫ x m ( a + b x n + c x 2 n ) p d x = − ( m − 3 n − 2 n p + 1 ) x m − 2 n + 1 ( a + b x n + c x 2 n ) p + 1 2 c n 2 ( p + 1 ) ( 2 p + 1 ) − x m − 2 n + 1 ( 2 a + b x n ) ( a + b x n + c x 2 n ) p 2 c n ( 2 p + 1 ) + ( m − n + 1 ) ( m − 2 n + 1 ) 2 c n 2 ( p + 1 ) ( 2 p + 1 ) ∫ x m − 2 n ( a + b x n + c x 2 n ) p + 1 d x ∫ x m ( a + b x n + c x 2 n ) p d x = x m + 1 ( a + b x n + c x 2 n ) p m + 2 n p + 1 + n p x m + 1 ( 2 a + b x n ) ( a + b x n + c x 2 n ) p − 1 ( m + 2 n p + 1 ) ( m + n ( 2 p − 1 ) + 1 ) + 2 a n 2 p ( 2 p − 1 ) ( m + 2 n p + 1 ) ( m + n ( 2 p − 1 ) + 1 ) ∫ x m ( a + b x n + c x 2 n ) p − 1 d x ∫ x m ( a + b x n + c x 2 n ) p d x = − ( m + n + 2 n p + 1 ) x m + 1 ( a + b x n + c x 2 n ) p + 1 2 a n 2 ( p + 1 ) ( 2 p + 1 ) − x m + 1 ( 2 a + b x n ) ( a + b x n + c x 2 n ) p 2 a n ( 2 p + 1 ) + ( m + n ( 2 p + 1 ) + 1 ) ( m + 2 n ( p + 1 ) + 1 ) 2 a n 2 ( p + 1 ) ( 2 p + 1 ) ∫ x m ( a + b x n + c x 2 n ) p + 1 d x ∫ x m ( a + b x n + c x 2 n ) p d x = x m − n + 1 ( b + 2 c x n ) ( a + b x n + c x 2 n ) p 2 c ( m + 2 n p + 1 ) − b ( m − n + 1 ) 2 c ( m + 2 n p + 1 ) ∫ x m − n ( a + b x n + c x 2 n ) p d x ∫ x m ( a + b x n + c x 2 n ) p d x = x m + 1 ( b + 2 c x n ) ( a + b x n + c x 2 n ) p b ( m + 1 ) − 2 c ( m + n ( 2 p + 1 ) + 1 ) b ( m + 1 ) ∫ x m + n ( a + b x n + c x 2 n ) p d x

Integrands of the form xm (A + B xn) (a + b xn + c x2n)p

The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.

These reduction formulas can be used for integrands having integer and/or fractional exponents.

Special cases of these reductions formulas can be used for integrands of the form ( a + b x n + c x 2 n ) p and x m ( a + b x n + c x 2 n ) p by setting m and/or B to 0.

∫ x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = x m + 1 ( A ( m + n ( 2 p + 1 ) + 1 ) + B ( m + 1 ) x n ) ( a + b x n + c x 2 n ) p ( m + 1 ) ( m + n ( 2 p + 1 ) + 1 ) + n p ( m + 1 ) ( m + n ( 2 p + 1 ) + 1 ) ⋅ ∫ x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = x m − n + 1 ( A b − 2 a B − ( b B − 2 A c ) x n ) ( a + b x n + c x 2 n ) p + 1 n ( p + 1 ) ( b 2 − 4 a c ) + 1 n ( p + 1 ) ( b 2 − 4 a c ) ⋅ ∫ x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = x m + 1 ( b B n p + A c ( m + n ( 2 p + 1 ) + 1 ) + B c ( m + 2 n p + 1 ) x n ) ( a + b x n + c x 2 n ) p c ( m + 2 n p + 1 ) ( m + n ( 2 p + 1 ) + 1 ) + n p c ( m + 2 n p + 1 ) ( m + n ( 2 p + 1 ) + 1 ) ⋅ ∫ x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = − x m + 1 ( A b 2 − a b B − 2 a A c + ( A b − 2 a B ) c x n ) ( a + b x n + c x 2 n ) p + 1 a n ( p + 1 ) ( b 2 − 4 a c ) + 1 a n ( p + 1 ) ( b 2 − 4 a c ) ⋅ ∫ x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = B x m − n + 1 ( a + b x n + c x 2 n ) p + 1 c ( m + n ( 2 p + 1 ) + 1 ) − 1 c ( m + n ( 2 p + 1 ) + 1 ) ⋅ ∫ x m ( A + B x n ) ( a + b x n + c x 2 n ) p d x = A x m + 1 ( a + b x n + c x 2 n ) p + 1 a ( m + 1 ) + 1 a ( m + 1 ) ⋅