Optimal. Leaf size=45 \[ -\frac {(c x)^{m-2} \, _2F_1\left (1,\frac {m-2}{2};\frac {m}{2};-\frac {b x^2}{a}\right )}{a c (2-m)} \]
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Rubi [A] time = 0.01, antiderivative size = 45, 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} \[ -\frac {(c x)^{m-2} \, _2F_1\left (1,\frac {m-2}{2};\frac {m}{2};-\frac {b x^2}{a}\right )}{a c (2-m)} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin {align*} \int \frac {(c x)^{-3+m}}{a+b x^2} \, dx &=-\frac {(c x)^{-2+m} \, _2F_1\left (1,\frac {1}{2} (-2+m);\frac {m}{2};-\frac {b x^2}{a}\right )}{a c (2-m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 0.98 \[ \frac {x (c x)^{m-3} \, _2F_1\left (1,\frac {m-2}{2};\frac {m-2}{2}+1;-\frac {b x^2}{a}\right )}{a (m-2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (c x\right )^{m - 3}}{b x^{2} + a}, x\right ) \]
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 \[ \int \frac {\left (c x\right )^{m - 3}}{b x^{2} + a}\,{d x} \]
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 \[ \int \frac {\left (c x \right )^{m -3}}{b \,x^{2}+a}\, dx \]
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 \[ \int \frac {\left (c x\right )^{m - 3}}{b x^{2} + a}\,{d x} \]
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 \[ \int \frac {{\left (c\,x\right )}^{m-3}}{b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 21.42, size = 92, normalized size = 2.04 \[ \frac {c^{m} m x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} - 1\right ) \Gamma \left (\frac {m}{2} - 1\right )}{4 a c^{3} x^{2} \Gamma \left (\frac {m}{2}\right )} - \frac {c^{m} x^{m} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} - 1\right ) \Gamma \left (\frac {m}{2} - 1\right )}{2 a c^{3} x^{2} \Gamma \left (\frac {m}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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