Optimal. Leaf size=85 \[ \frac{a^2 x \sqrt [3]{a+b x^3} (11 b c-a d) \, _2F_1\left (-\frac{7}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{11 b \sqrt [3]{\frac{b x^3}{a}+1}}+\frac{d x \left (a+b x^3\right )^{10/3}}{11 b} \]
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Rubi [A] time = 0.0228746, antiderivative size = 85, 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, 246, 245} \[ \frac{a^2 x \sqrt [3]{a+b x^3} (11 b c-a d) \, _2F_1\left (-\frac{7}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{11 b \sqrt [3]{\frac{b x^3}{a}+1}}+\frac{d x \left (a+b x^3\right )^{10/3}}{11 b} \]
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 )^{7/3} \left (c+d x^3\right ) \, dx &=\frac{d x \left (a+b x^3\right )^{10/3}}{11 b}-\frac{(-11 b c+a d) \int \left (a+b x^3\right )^{7/3} \, dx}{11 b}\\ &=\frac{d x \left (a+b x^3\right )^{10/3}}{11 b}-\frac{\left (a^2 (-11 b c+a d) \sqrt [3]{a+b x^3}\right ) \int \left (1+\frac{b x^3}{a}\right )^{7/3} \, dx}{11 b \sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{d x \left (a+b x^3\right )^{10/3}}{11 b}+\frac{a^2 (11 b c-a d) x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{7}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{11 b \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [A] time = 0.037149, size = 77, normalized size = 0.91 \[ \frac{x \sqrt [3]{a+b x^3} \left (d \left (a+b x^3\right )^3-\frac{a^2 (a d-11 b c) \, _2F_1\left (-\frac{7}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\sqrt [3]{\frac{b x^3}{a}+1}}\right )}{11 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.22, size = 0, normalized size = 0. \begin{align*} \int \left ( b{x}^{3}+a \right ) ^{{\frac{7}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{3} + a\right )}^{\frac{7}{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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} d x^{9} +{\left (b^{2} c + 2 \, a b d\right )} x^{6} +{\left (2 \, a b c + a^{2} d\right )} x^{3} + a^{2} c\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 7.0308, size = 265, normalized size = 3.12 \begin{align*} \frac{a^{\frac{7}{3}} c x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{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{7}{3}} d x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{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 )} + \frac{2 a^{\frac{4}{3}} b c x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{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 )} + \frac{2 a^{\frac{4}{3}} b d x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} + \frac{\sqrt [3]{a} b^{2} c x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} + \frac{\sqrt [3]{a} b^{2} d x^{10} \Gamma \left (\frac{10}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{10}{3} \\ \frac{13}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{13}{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{7}{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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