Optimal. Leaf size=76 \[ \frac{(c x)^{m+1} (A b-a C) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b c (m+1)}+\frac{C (c x)^{m+1}}{b c (m+1)} \]
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Rubi [A] time = 0.0386468, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {459, 364} \[ \frac{(c x)^{m+1} (A b-a C) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a b c (m+1)}+\frac{C (c x)^{m+1}}{b c (m+1)} \]
Antiderivative was successfully verified.
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Rule 459
Rule 364
Rubi steps
\begin{align*} \int \frac{(c x)^m \left (A+C x^2\right )}{a+b x^2} \, dx &=\frac{C (c x)^{1+m}}{b c (1+m)}-\frac{(-A b (1+m)+a C (1+m)) \int \frac{(c x)^m}{a+b x^2} \, dx}{b (1+m)}\\ &=\frac{C (c x)^{1+m}}{b c (1+m)}+\frac{(A b-a C) (c x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{b x^2}{a}\right )}{a b c (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0537702, size = 56, normalized size = 0.74 \[ \frac{x (c x)^m \left ((A b-a C) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )+a C\right )}{a b (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( C{x}^{2}+A \right ) \left ( cx \right ) ^{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^{2} + A\right )} \left (c x\right )^{m}}{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^{2} + A\right )} \left (c x\right )^{m}}{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 = 5.37861, size = 204, normalized size = 2.68 \begin{align*} \frac{A c^{m} 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{A c^{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{C c^{m} m x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{2}\right )} + \frac{3 C c^{m} x^{3} x^{m} \Phi \left (\frac{b x^{2} e^{i \pi }}{a}, 1, \frac{m}{2} + \frac{3}{2}\right ) \Gamma \left (\frac{m}{2} + \frac{3}{2}\right )}{4 a \Gamma \left (\frac{m}{2} + \frac{5}{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^{2} + A\right )} \left (c x\right )^{m}}{b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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