Integrand size = 41, antiderivative size = 36 \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {a x^2}{b}\right )}{1+m} \]
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Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {74, 371} \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{2},\frac {m+3}{2},-\frac {a x^2}{b}\right )}{m+1} \]
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Rule 74
Rule 371
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m}{\left (1+\frac {a x^2}{b}\right )^2} \, dx \\ & = \frac {x^{1+m} \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {a x^2}{b}\right )}{1+m} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},1+\frac {1+m}{2},-\frac {a x^2}{b}\right )}{1+m} \]
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\[\int \frac {x^{m}}{\left (1-\frac {x \sqrt {a}}{\sqrt {-b}}\right )^{2} \left (1+\frac {x \sqrt {a}}{\sqrt {-b}}\right )^{2}}d x\]
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\[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\int { \frac {x^{m}}{{\left (\frac {\sqrt {a} x}{\sqrt {-b}} + 1\right )}^{2} {\left (\frac {\sqrt {a} x}{\sqrt {-b}} - 1\right )}^{2}} \,d x } \]
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Result contains complex when optimal does not.
Time = 3.66 (sec) , antiderivative size = 552, normalized size of antiderivative = 15.33 \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\frac {a b^{2} m^{2} x^{2} x^{m - 3} \Phi \left (\frac {b e^{i \pi }}{a x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} - \frac {4 a b^{2} m x^{2} x^{m - 3} \Phi \left (\frac {b e^{i \pi }}{a x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} + \frac {2 a b^{2} m x^{2} x^{m - 3} \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} + \frac {3 a b^{2} x^{2} x^{m - 3} \Phi \left (\frac {b e^{i \pi }}{a x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} - \frac {6 a b^{2} x^{2} x^{m - 3} \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} + \frac {b^{3} m^{2} x^{m - 3} \Phi \left (\frac {b e^{i \pi }}{a x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} - \frac {4 b^{3} m x^{m - 3} \Phi \left (\frac {b e^{i \pi }}{a x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} + \frac {3 b^{3} x^{m - 3} \Phi \left (\frac {b e^{i \pi }}{a x^{2}}, 1, \frac {3}{2} - \frac {m}{2}\right ) \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{8 a^{3} x^{2} \Gamma \left (\frac {5}{2} - \frac {m}{2}\right ) + 8 a^{2} b \Gamma \left (\frac {5}{2} - \frac {m}{2}\right )} \]
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\[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\int { \frac {x^{m}}{{\left (\frac {\sqrt {a} x}{\sqrt {-b}} + 1\right )}^{2} {\left (\frac {\sqrt {a} x}{\sqrt {-b}} - 1\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^m}{\left (1-\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2 \left (1+\frac {\sqrt {a} x}{\sqrt {-b}}\right )^2} \, dx=\int \frac {x^m}{{\left (\frac {\sqrt {a}\,x}{\sqrt {-b}}-1\right )}^2\,{\left (\frac {\sqrt {a}\,x}{\sqrt {-b}}+1\right )}^2} \,d x \]
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