Integrand size = 19, antiderivative size = 70 \[ \int \frac {x^m}{(1-a x)^2 (1+a x)} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{1+m}+\frac {a x^{2+m} \operatorname {Hypergeometric2F1}\left (2,\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{2+m} \]
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Time = 0.01 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {83, 74, 371} \[ \int \frac {x^m}{(1-a x)^2 (1+a x)} \, dx=\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{2},\frac {m+3}{2},a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \operatorname {Hypergeometric2F1}\left (2,\frac {m+2}{2},\frac {m+4}{2},a^2 x^2\right )}{m+2} \]
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Rule 74
Rule 83
Rule 371
Rubi steps \begin{align*} \text {integral}& = a \int \frac {x^{1+m}}{(1-a x)^2 (1+a x)^2} \, dx+\int \frac {x^m}{(1-a x)^2 (1+a x)^2} \, dx \\ & = a \int \frac {x^{1+m}}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {x^m}{\left (1-a^2 x^2\right )^2} \, dx \\ & = \frac {x^{1+m} \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};a^2 x^2\right )}{1+m}+\frac {a x^{2+m} \, _2F_1\left (2,\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{2+m} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.96 \[ \int \frac {x^m}{(1-a x)^2 (1+a x)} \, dx=x^{1+m} \left (\frac {a x \operatorname {Hypergeometric2F1}\left (2,1+\frac {m}{2},2+\frac {m}{2},a^2 x^2\right )}{2+m}+\frac {\operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{1+m}\right ) \]
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\[\int \frac {x^{m}}{\left (-a x +1\right )^{2} \left (a x +1\right )}d x\]
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\[ \int \frac {x^m}{(1-a x)^2 (1+a x)} \, dx=\int { \frac {x^{m}}{{\left (a x + 1\right )} {\left (a x - 1\right )}^{2}} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.35 (sec) , antiderivative size = 313, normalized size of antiderivative = 4.47 \[ \int \frac {x^m}{(1-a x)^2 (1+a x)} \, dx=\frac {2 a m^{2} x x^{m} \Phi \left (\frac {1}{a x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{4 a^{2} x \Gamma \left (1 - m\right ) - 4 a \Gamma \left (1 - m\right )} - \frac {a m x x^{m} \Phi \left (\frac {1}{a x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{4 a^{2} x \Gamma \left (1 - m\right ) - 4 a \Gamma \left (1 - m\right )} + \frac {a m x x^{m} \Phi \left (\frac {e^{i \pi }}{a x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{4 a^{2} x \Gamma \left (1 - m\right ) - 4 a \Gamma \left (1 - m\right )} + \frac {2 a m x x^{m} \Gamma \left (- m\right )}{4 a^{2} x \Gamma \left (1 - m\right ) - 4 a \Gamma \left (1 - m\right )} - \frac {2 m^{2} x^{m} \Phi \left (\frac {1}{a x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{4 a^{2} x \Gamma \left (1 - m\right ) - 4 a \Gamma \left (1 - m\right )} + \frac {m x^{m} \Phi \left (\frac {1}{a x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{4 a^{2} x \Gamma \left (1 - m\right ) - 4 a \Gamma \left (1 - m\right )} - \frac {m x^{m} \Phi \left (\frac {e^{i \pi }}{a x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{4 a^{2} x \Gamma \left (1 - m\right ) - 4 a \Gamma \left (1 - m\right )} \]
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\[ \int \frac {x^m}{(1-a x)^2 (1+a x)} \, dx=\int { \frac {x^{m}}{{\left (a x + 1\right )} {\left (a x - 1\right )}^{2}} \,d x } \]
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\[ \int \frac {x^m}{(1-a x)^2 (1+a x)} \, dx=\int { \frac {x^{m}}{{\left (a x + 1\right )} {\left (a x - 1\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^m}{(1-a x)^2 (1+a x)} \, dx=\int \frac {x^m}{{\left (a\,x-1\right )}^2\,\left (a\,x+1\right )} \,d x \]
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