有理関数の原始関数の一覧
∫ 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 {\displaystyle \int {\frac {x}{(ax^{2}+bx+c)^{n}}}dx=-{\frac {bx+2c}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}}}-{\frac {b(2n-3)}{(n-1)(4ac-b^{2})}}\int {\frac {1}{(ax^{2}+bx+c)^{n-1}}}dx+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 {\displaystyle \int {\frac {1}{x(ax^{2}+bx+c)}}dx={\frac {1}{2c}}\ln \left|{\frac {x^{2}}{ax^{2}+bx+c}}\right|-{\frac {b}{2c}}\int {\frac {1}{ax^{2}+bx+c}}dx+C}