Optimal. Leaf size=44 \[ \frac{(c x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right )}{a c (m+2)} \]
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Rubi [A] time = 0.0118202, antiderivative size = 44, normalized size of antiderivative = 1., 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} \[ \frac{(c x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right )}{a c (m+2)} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin{align*} \int \frac{(c x)^{1+m}}{a+b x^2} \, dx &=\frac{(c x)^{2+m} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};-\frac{b x^2}{a}\right )}{a c (2+m)}\\ \end{align*}
Mathematica [A] time = 0.0144331, size = 45, normalized size = 1.02 \[ \frac{c x^2 (c x)^m \, _2F_1\left (1,\frac{m+2}{2};\frac{m+2}{2}+1;-\frac{b x^2}{a}\right )}{a (m+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( cx \right ) ^{1+m}}{b{x}^{2}+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{m + 1}}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (c x\right )^{m + 1}}{b x^{2} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 5.3791, size = 92, normalized size = 2.09 \begin{align*} \frac{c c^{m} m x^{2} 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 \Gamma \left (\frac{m}{2} + 2\right )} + \frac{c c^{m} x^{2} 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 \Gamma \left (\frac{m}{2} + 2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{m + 1}}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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