Optimal. Leaf size=66 \[ \frac{(c x)^{m+1} \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{3},-p;\frac{m+4}{3};-\frac{b x^3}{a}\right )}{c (m+1)} \]
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Rubi [A] time = 0.0176311, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {365, 364} \[ \frac{(c x)^{m+1} \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{3},-p;\frac{m+4}{3};-\frac{b x^3}{a}\right )}{c (m+1)} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int (c x)^m \left (a+b x^3\right )^p \, dx &=\left (\left (a+b x^3\right )^p \left (1+\frac{b x^3}{a}\right )^{-p}\right ) \int (c x)^m \left (1+\frac{b x^3}{a}\right )^p \, dx\\ &=\frac{(c x)^{1+m} \left (a+b x^3\right )^p \left (1+\frac{b x^3}{a}\right )^{-p} \, _2F_1\left (\frac{1+m}{3},-p;\frac{4+m}{3};-\frac{b x^3}{a}\right )}{c (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0134546, size = 64, normalized size = 0.97 \[ \frac{x (c x)^m \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{3},-p;\frac{m+1}{3}+1;-\frac{b x^3}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{m} \left ( b{x}^{3}+a \right ) ^{p}\, 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{\left (b x^{3} + a\right )}^{p} \left (c x\right )^{m}\,{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 ({\left (b x^{3} + a\right )}^{p} \left (c x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (b x^{3} + a\right )}^{p} \left (c x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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