Optimal. Leaf size=141 \[ -\frac{a (6 b c-a d) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 b^{4/3}}+\frac{a (6 b c-a d) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{4/3}}+\frac{x \left (a+b x^3\right )^{2/3} (6 b c-a d)}{18 b}+\frac{d x \left (a+b x^3\right )^{5/3}}{6 b} \]
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Rubi [A] time = 0.0450737, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {388, 195, 239} \[ -\frac{a (6 b c-a d) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 b^{4/3}}+\frac{a (6 b c-a d) \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{4/3}}+\frac{x \left (a+b x^3\right )^{2/3} (6 b c-a d)}{18 b}+\frac{d x \left (a+b x^3\right )^{5/3}}{6 b} \]
Antiderivative was successfully verified.
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Rule 388
Rule 195
Rule 239
Rubi steps
\begin{align*} \int \left (a+b x^3\right )^{2/3} \left (c+d x^3\right ) \, dx &=\frac{d x \left (a+b x^3\right )^{5/3}}{6 b}-\frac{(-6 b c+a d) \int \left (a+b x^3\right )^{2/3} \, dx}{6 b}\\ &=\frac{(6 b c-a d) x \left (a+b x^3\right )^{2/3}}{18 b}+\frac{d x \left (a+b x^3\right )^{5/3}}{6 b}+\frac{(a (6 b c-a d)) \int \frac{1}{\sqrt [3]{a+b x^3}} \, dx}{9 b}\\ &=\frac{(6 b c-a d) x \left (a+b x^3\right )^{2/3}}{18 b}+\frac{d x \left (a+b x^3\right )^{5/3}}{6 b}+\frac{a (6 b c-a d) \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{9 \sqrt{3} b^{4/3}}-\frac{a (6 b c-a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 b^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.0694077, size = 72, normalized size = 0.51 \[ \frac{x \left (a+b x^3\right )^{2/3} \left (\frac{(6 b c-a d) \, _2F_1\left (-\frac{2}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (\frac{b x^3}{a}+1\right )^{2/3}}+d \left (a+b x^3\right )\right )}{6 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.222, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02218, size = 1087, normalized size = 7.71 \begin{align*} \left [-\frac{3 \, \sqrt{\frac{1}{3}}{\left (6 \, a b^{2} c - a^{2} b d\right )} \sqrt{-\frac{1}{b^{\frac{2}{3}}}} \log \left (3 \, b x^{3} - 3 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}} x^{2} - 3 \, \sqrt{\frac{1}{3}}{\left (b^{\frac{4}{3}} x^{3} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} b x^{2} - 2 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b^{\frac{2}{3}} x\right )} \sqrt{-\frac{1}{b^{\frac{2}{3}}}} + 2 \, a\right ) + 2 \,{\left (6 \, a b c - a^{2} d\right )} b^{\frac{2}{3}} \log \left (-\frac{b^{\frac{1}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) -{\left (6 \, a b c - a^{2} d\right )} b^{\frac{2}{3}} \log \left (\frac{b^{\frac{2}{3}} x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right ) - 3 \,{\left (3 \, b^{2} d x^{4} + 2 \,{\left (3 \, b^{2} c + a b d\right )} x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{54 \, b^{2}}, -\frac{2 \,{\left (6 \, a b c - a^{2} d\right )} b^{\frac{2}{3}} \log \left (-\frac{b^{\frac{1}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) -{\left (6 \, a b c - a^{2} d\right )} b^{\frac{2}{3}} \log \left (\frac{b^{\frac{2}{3}} x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right ) + \frac{6 \, \sqrt{\frac{1}{3}}{\left (6 \, a b^{2} c - a^{2} b d\right )} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (b^{\frac{1}{3}} x + 2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}}\right )}}{b^{\frac{1}{3}} x}\right )}{b^{\frac{1}{3}}} - 3 \,{\left (3 \, b^{2} d x^{4} + 2 \,{\left (3 \, b^{2} c + a b d\right )} x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{54 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.2304, size = 82, normalized size = 0.58 \begin{align*} \frac{a^{\frac{2}{3}} c x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} + \frac{a^{\frac{2}{3}} d x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\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{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (d x^{3} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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