Optimal. Leaf size=47 \[ -\frac {(c x)^{m-1} \, _2F_1\left (1,\frac {m-1}{2};\frac {m+1}{2};-\frac {b x^2}{a}\right )}{a c (1-m)} \]
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Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {364} \[ -\frac {(c x)^{m-1} \, _2F_1\left (1,\frac {m-1}{2};\frac {m+1}{2};-\frac {b x^2}{a}\right )}{a c (1-m)} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin {align*} \int \frac {(c x)^{-2+m}}{a+b x^2} \, dx &=-\frac {(c x)^{-1+m} \, _2F_1\left (1,\frac {1}{2} (-1+m);\frac {1+m}{2};-\frac {b x^2}{a}\right )}{a c (1-m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 0.94 \[ \frac {x (c x)^{m-2} \, _2F_1\left (1,\frac {m-1}{2};\frac {m-1}{2}+1;-\frac {b x^2}{a}\right )}{a (m-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (c x\right )^{m - 2}}{b x^{2} + a}, x\right ) \]
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 \[ \int \frac {\left (c x\right )^{m - 2}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{m -2}}{b \,x^{2}+a}\, dx \]
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 \[ \int \frac {\left (c x\right )^{m - 2}}{b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c\,x\right )}^{m-2}}{b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 11.57, size = 102, normalized size = 2.17 \[ \frac {c^{m} m x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} - \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} - \frac {1}{2}\right )}{4 a c^{2} x \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )} - \frac {c^{m} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} - \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} - \frac {1}{2}\right )}{4 a c^{2} x \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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