Optimal. Leaf size=93 \[ \frac{d x \left (a+b x^3\right )^{m+1}}{b (3 m+4)}-\frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} (a d-b c (3 m+4)) \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )}{b (3 m+4)} \]
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Rubi [A] time = 0.0424879, antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {388, 246, 245} \[ x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left (c-\frac{a d}{3 b m+4 b}\right ) \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )+\frac{d x \left (a+b x^3\right )^{m+1}}{b (3 m+4)} \]
Antiderivative was successfully verified.
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Rule 388
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a+b x^3\right )^m \left (c+d x^3\right ) \, dx &=\frac{d x \left (a+b x^3\right )^{1+m}}{b (4+3 m)}-\left (-c+\frac{a d}{4 b+3 b m}\right ) \int \left (a+b x^3\right )^m \, dx\\ &=\frac{d x \left (a+b x^3\right )^{1+m}}{b (4+3 m)}-\left (\left (-c+\frac{a d}{4 b+3 b m}\right ) \left (a+b x^3\right )^m \left (1+\frac{b x^3}{a}\right )^{-m}\right ) \int \left (1+\frac{b x^3}{a}\right )^m \, dx\\ &=\frac{d x \left (a+b x^3\right )^{1+m}}{b (4+3 m)}+\left (c-\frac{a d}{4 b+3 b m}\right ) x \left (a+b x^3\right )^m \left (1+\frac{b x^3}{a}\right )^{-m} \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0308868, size = 90, normalized size = 0.97 \[ \frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} \left ((b c (3 m+4)-a d) \, _2F_1\left (\frac{1}{3},-m;\frac{4}{3};-\frac{b x^3}{a}\right )+d \left (a+b x^3\right ) \left (\frac{b x^3}{a}+1\right )^m\right )}{b (3 m+4)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.225, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{3}+a \right ) ^{m} \left ( d{x}^{3}+c \right ) \, 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 (d x^{3} + c\right )}{\left (b x^{3} + a\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 (d x^{3} + c\right )}{\left (b x^{3} + a\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 (d x^{3} + c\right )}{\left (b x^{3} + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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