Optimal. Leaf size=36 \[ \frac {x^{1+m} \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {a x^2}{b}\right )}{1+m} \]
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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {74, 371}
\begin {gather*} \frac {x^{m+1} \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};-\frac {a x^2}{b}\right )}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rule 371
Rubi steps
\begin {align*} \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 (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*}
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Mathematica [A]
time = 0.01, size = 38, normalized size = 1.06 \begin {gather*} \frac {x^{1+m} \, _2F_1\left (2,\frac {1+m}{2};1+\frac {1+m}{2};-\frac {a x^2}{b}\right )}{1+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\left (1-\frac {x \sqrt {a}}{\sqrt {-b}}\right )^{2} \left (1+\frac {x \sqrt {a}}{\sqrt {-b}}\right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 3.62, size = 541, normalized size = 15.03 \begin {gather*} \frac {a b^{2} m^{2} x^{m} \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 )}{x \left (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 )\right )} - \frac {4 a b^{2} m x^{m} \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 )}{x \left (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 )\right )} + \frac {2 a b^{2} m x^{m} \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{x \left (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 )\right )} + \frac {3 a b^{2} x^{m} \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 )}{x \left (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 )\right )} - \frac {6 a b^{2} x^{m} \Gamma \left (\frac {3}{2} - \frac {m}{2}\right )}{x \left (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 )\right )} + \frac {b^{3} m^{2} x^{m} \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 )}{x^{3} \cdot \left (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 )\right )} - \frac {4 b^{3} m x^{m} \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 )}{x^{3} \cdot \left (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 )\right )} + \frac {3 b^{3} x^{m} \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 )}{x^{3} \cdot \left (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 )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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