Optimal. Leaf size=94 \[ \frac{57 a^3 x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{44 \sqrt [3]{\frac{b x^3}{a}+1}}-\frac{9}{44} a x \left (a+b x^3\right )^{7/3}-\frac{1}{11} x \left (a-b x^3\right ) \left (a+b x^3\right )^{7/3} \]
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Rubi [A] time = 0.033539, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {416, 388, 246, 245} \[ \frac{57 a^3 x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{44 \sqrt [3]{\frac{b x^3}{a}+1}}-\frac{9}{44} a x \left (a+b x^3\right )^{7/3}-\frac{1}{11} x \left (a-b x^3\right ) \left (a+b x^3\right )^{7/3} \]
Antiderivative was successfully verified.
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Rule 416
Rule 388
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{4/3} \, dx &=-\frac{1}{11} x \left (a-b x^3\right ) \left (a+b x^3\right )^{7/3}+\frac{\int \left (a+b x^3\right )^{4/3} \left (12 a^2 b-18 a b^2 x^3\right ) \, dx}{11 b}\\ &=-\frac{9}{44} a x \left (a+b x^3\right )^{7/3}-\frac{1}{11} x \left (a-b x^3\right ) \left (a+b x^3\right )^{7/3}+\frac{1}{44} \left (57 a^2\right ) \int \left (a+b x^3\right )^{4/3} \, dx\\ &=-\frac{9}{44} a x \left (a+b x^3\right )^{7/3}-\frac{1}{11} x \left (a-b x^3\right ) \left (a+b x^3\right )^{7/3}+\frac{\left (57 a^3 \sqrt [3]{a+b x^3}\right ) \int \left (1+\frac{b x^3}{a}\right )^{4/3} \, dx}{44 \sqrt [3]{1+\frac{b x^3}{a}}}\\ &=-\frac{9}{44} a x \left (a+b x^3\right )^{7/3}-\frac{1}{11} x \left (a-b x^3\right ) \left (a+b x^3\right )^{7/3}+\frac{57 a^3 x \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{4}{3},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{44 \sqrt [3]{1+\frac{b x^3}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0481449, size = 97, normalized size = 1.03 \[ \frac{x \left (-78 a^2 b^2 x^6+114 a^4 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )+53 a^3 b x^3+106 a^4-5 a b^3 x^9+20 b^4 x^{12}\right )}{220 \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.215, size = 0, normalized size = 0. \begin{align*} \int \left ( -b{x}^{3}+a \right ) ^{2} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}\, 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{4}{3}}{\left (b x^{3} - a\right )}^{2}\,{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^{3} x^{9} - a b^{2} x^{6} - a^{2} b x^{3} + a^{3}\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 = 5.11501, size = 168, normalized size = 1.79 \begin{align*} \frac{a^{\frac{10}{3}} 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}} b 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{a^{\frac{4}{3}} b^{2} 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^{3} 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{4}{3}}{\left (b x^{3} - a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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