Optimal. Leaf size=36 \[ \frac {x^{m+1} \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};-\frac {a x^2}{b}\right )}{m+1} \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {364} \[ \frac {x^{m+1} \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};-\frac {a x^2}{b}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin {align*} \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 \[ \frac {x^{m+1} \, _2F_1\left (2,\frac {m+1}{2};\frac {m+1}{2}+1;-\frac {a x^2}{b}\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x^{m}}{a^{2} x^{4} + 2 \, a b x^{2} + b^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{{\left (\frac {a x^{2}}{b} + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 92, normalized size = 2.56 \[ \frac {\left (\frac {2 \left (-\frac {m^{2}}{4}+\frac {1}{4}\right ) x^{m +1} \left (\frac {a}{b}\right )^{\frac {m}{2}+\frac {1}{2}} \Phi \left (-\frac {a \,x^{2}}{b}, 1, \frac {m}{2}+\frac {1}{2}\right )}{m +1}+\frac {2 x^{m +1} \left (\frac {a}{b}\right )^{\frac {m}{2}+\frac {1}{2}}}{\frac {2 a \,x^{2}}{b}+2}\right ) \left (\frac {a}{b}\right )^{-\frac {m}{2}-\frac {1}{2}}}{2} \]
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 \[ \int \frac {x^{m}}{{\left (\frac {a x^{2}}{b} + 1\right )}^{2}}\,{d x} \]
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 \[ \int \frac {x^m}{{\left (\frac {a\,x^2}{b}+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.16, size = 343, normalized size = 9.53 \[ - \frac {a m^{2} x^{3} x^{m} \Phi \left (\frac {a x^{2} e^{i \pi }}{b}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 b \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {a x^{3} x^{m} \Phi \left (\frac {a x^{2} e^{i \pi }}{b}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 b \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} - \frac {b m^{2} x x^{m} \Phi \left (\frac {a x^{2} e^{i \pi }}{b}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 b \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {2 b m x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 b \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {b x x^{m} \Phi \left (\frac {a x^{2} e^{i \pi }}{b}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 b \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {2 b x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 a x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 b \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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