Optimal. Leaf size=39 \[ \frac {x^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a (m+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {364} \[ \frac {x^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{a (m+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin {align*} \int \frac {x^m}{a+b x^2} \, dx &=\frac {x^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 1.05 \[ \frac {x^{m+1} \, _2F_1\left (1,\frac {m+1}{2};\frac {m+1}{2}+1;-\frac {b x^2}{a}\right )}{a (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{b x^{2} + a}, x\right ) \]
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 \[ \int \frac {x^{m}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{b \,x^{2}+a}\, dx \]
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 \[ \int \frac {x^{m}}{b x^{2} + a}\,{d x} \]
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 \[ \int \frac {x^m}{b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.25, size = 88, normalized size = 2.26 \[ \frac {m x x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {x x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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