Optimal. Leaf size=66 \[ \frac{x^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a b (m+1)}+\frac{B x^{m+1}}{b (m+1)} \]
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Rubi [A] time = 0.0350864, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {459, 364} \[ \frac{x^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a b (m+1)}+\frac{B x^{m+1}}{b (m+1)} \]
Antiderivative was successfully verified.
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Rule 459
Rule 364
Rubi steps
\begin{align*} \int \frac{x^m \left (A+B x^3\right )}{a+b x^3} \, dx &=\frac{B x^{1+m}}{b (1+m)}-\frac{(-A b (1+m)+a B (1+m)) \int \frac{x^m}{a+b x^3} \, dx}{b (1+m)}\\ &=\frac{B x^{1+m}}{b (1+m)}+\frac{(A b-a B) x^{1+m} \, _2F_1\left (1,\frac{1+m}{3};\frac{4+m}{3};-\frac{b x^3}{a}\right )}{a b (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0587332, size = 55, normalized size = 0.83 \[ \frac{x^{m+1} \left ((A b-a B) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )+a B\right )}{a b (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.104, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m} \left ( B{x}^{3}+A \right ) }{b{x}^{3}+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 (B x^{3} + A\right )} x^{m}}{b x^{3} + 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 (B x^{3} + A\right )} x^{m}}{b x^{3} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 26.9433, size = 190, normalized size = 2.88 \begin{align*} \frac{A m x x^{m} \Phi \left (\frac{b x^{3} e^{i \pi }}{a}, 1, \frac{m}{3} + \frac{1}{3}\right ) \Gamma \left (\frac{m}{3} + \frac{1}{3}\right )}{9 a \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} + \frac{A x x^{m} \Phi \left (\frac{b x^{3} e^{i \pi }}{a}, 1, \frac{m}{3} + \frac{1}{3}\right ) \Gamma \left (\frac{m}{3} + \frac{1}{3}\right )}{9 a \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} + \frac{B m x^{4} x^{m} \Phi \left (\frac{b x^{3} e^{i \pi }}{a}, 1, \frac{m}{3} + \frac{4}{3}\right ) \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )}{9 a \Gamma \left (\frac{m}{3} + \frac{7}{3}\right )} + \frac{4 B x^{4} x^{m} \Phi \left (\frac{b x^{3} e^{i \pi }}{a}, 1, \frac{m}{3} + \frac{4}{3}\right ) \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )}{9 a \Gamma \left (\frac{m}{3} + \frac{7}{3}\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 (B x^{3} + A\right )} x^{m}}{b x^{3} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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