Optimal. Leaf size=82 \[ \frac{x \sqrt [3]{a+b x^3} (5 b c-a d) \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{5 b \sqrt [3]{\frac{b x^3}{a}+1}}+\frac{d x \left (a+b x^3\right )^{4/3}}{5 b} \]
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Rubi [A] time = 0.0210526, antiderivative size = 82, 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} \[ \frac{x \sqrt [3]{a+b x^3} (5 b c-a d) \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{5 b \sqrt [3]{\frac{b x^3}{a}+1}}+\frac{d x \left (a+b x^3\right )^{4/3}}{5 b} \]
Antiderivative was successfully verified.
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Rule 388
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \sqrt [3]{a+b x^3} \left (c+d x^3\right ) \, dx &=\frac{d x \left (a+b x^3\right )^{4/3}}{5 b}-\frac{(-5 b c+a d) \int \sqrt [3]{a+b x^3} \, dx}{5 b}\\ &=\frac{d x \left (a+b x^3\right )^{4/3}}{5 b}-\frac{\left ((-5 b c+a d) \sqrt [3]{a+b x^3}\right ) \int \sqrt [3]{1+\frac{b x^3}{a}} \, dx}{5 b \sqrt [3]{1+\frac{b x^3}{a}}}\\ &=\frac{d x \left (a+b x^3\right )^{4/3}}{5 b}+\frac{(5 b c-a d) x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{5 b \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0669086, size = 72, normalized size = 0.88 \[ \frac{x \sqrt [3]{a+b x^3} \left (\frac{(5 b c-a d) \, _2F_1\left (-\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\sqrt [3]{\frac{b x^3}{a}+1}}+d \left (a+b x^3\right )\right )}{5 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.208, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{b{x}^{3}+a} \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{1}{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 x^{3} + a\right )}^{\frac{1}{3}}{\left (d x^{3} + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.0371, size = 82, normalized size = 1. \begin{align*} \frac{\sqrt [3]{a} 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{\sqrt [3]{a} 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 )} \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{1}{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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