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 = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {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 371
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 = 0.03, 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 [C] Result contains higher order function than in optimal. Order 9 vs. order
5.
time = 0.06, size = 92, normalized size = 2.56
method | result | size |
meijerg | \(\frac {\left (\frac {a}{b}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {2 x^{1+m} \left (\frac {a}{b}\right )^{\frac {1}{2}+\frac {m}{2}}}{2+\frac {2 a \,x^{2}}{b}}+\frac {2 x^{1+m} \left (\frac {a}{b}\right )^{\frac {1}{2}+\frac {m}{2}} \left (-\frac {m^{2}}{4}+\frac {1}{4}\right ) \Phi \left (-\frac {a \,x^{2}}{b}, 1, \frac {1}{2}+\frac {m}{2}\right )}{1+m}\right )}{2}\) | \(92\) |
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 = 3.01, size = 343, normalized size = 9.53 \begin {gather*} - \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 )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {a\,x^2}{b}+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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