Optimal. Leaf size=139 \[ \frac{38}{81} a^2 x \left (a+b x^3\right )^{2/3}-\frac{38 a^3 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{81 \sqrt [3]{b}}+\frac{76 a^3 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{81 \sqrt{3} \sqrt [3]{b}}-\frac{8}{27} a x \left (a+b x^3\right )^{5/3}-\frac{1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3} \]
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Rubi [A] time = 0.0566985, antiderivative size = 139, 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, 195, 239} \[ \frac{38}{81} a^2 x \left (a+b x^3\right )^{2/3}-\frac{38 a^3 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{81 \sqrt [3]{b}}+\frac{76 a^3 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{81 \sqrt{3} \sqrt [3]{b}}-\frac{8}{27} a x \left (a+b x^3\right )^{5/3}-\frac{1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3} \]
Antiderivative was successfully verified.
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Rule 416
Rule 388
Rule 195
Rule 239
Rubi steps
\begin{align*} \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{2/3} \, dx &=-\frac{1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}+\frac{\int \left (a+b x^3\right )^{2/3} \left (10 a^2 b-16 a b^2 x^3\right ) \, dx}{9 b}\\ &=-\frac{8}{27} a x \left (a+b x^3\right )^{5/3}-\frac{1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}+\frac{1}{27} \left (38 a^2\right ) \int \left (a+b x^3\right )^{2/3} \, dx\\ &=\frac{38}{81} a^2 x \left (a+b x^3\right )^{2/3}-\frac{8}{27} a x \left (a+b x^3\right )^{5/3}-\frac{1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}+\frac{1}{81} \left (76 a^3\right ) \int \frac{1}{\sqrt [3]{a+b x^3}} \, dx\\ &=\frac{38}{81} a^2 x \left (a+b x^3\right )^{2/3}-\frac{8}{27} a x \left (a+b x^3\right )^{5/3}-\frac{1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}+\frac{76 a^3 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{81 \sqrt{3} \sqrt [3]{b}}-\frac{38 a^3 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{81 \sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.137231, size = 151, normalized size = 1.09 \[ \frac{1}{243} \left (3 \left (a+b x^3\right )^{2/3} \left (5 a^2 x-24 a b x^4+9 b^2 x^7\right )+\frac{38 a^3 \left (\log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )\right )}{\sqrt [3]{b}}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.233, size = 0, normalized size = 0. \begin{align*} \int \left ( -b{x}^{3}+a \right ) ^{2} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}\, 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 = 2.63047, size = 1076, normalized size = 7.74 \begin{align*} \left [\frac{114 \, \sqrt{\frac{1}{3}} a^{3} b \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{2}{3}} x^{2} - 3 \, \sqrt{\frac{1}{3}}{\left (\left (-b\right )^{\frac{1}{3}} b x^{3} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b x^{2} + 2 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-b\right )^{\frac{2}{3}} x\right )} \sqrt{\frac{\left (-b\right )^{\frac{1}{3}}}{b}} + 2 \, a\right ) - 76 \, a^{3} \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) + 38 \, a^{3} \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{2}{3}} x^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right ) + 3 \,{\left (9 \, b^{3} x^{7} - 24 \, a b^{2} x^{4} + 5 \, a^{2} b x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{243 \, b}, -\frac{228 \, \sqrt{\frac{1}{3}} a^{3} b \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}} \arctan \left (-\frac{\sqrt{\frac{1}{3}}{\left (\left (-b\right )^{\frac{1}{3}} x - 2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-b\right )^{\frac{1}{3}}}{b}}}{x}\right ) + 76 \, a^{3} \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) - 38 \, a^{3} \left (-b\right )^{\frac{2}{3}} \log \left (\frac{\left (-b\right )^{\frac{2}{3}} x^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b\right )^{\frac{1}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{2}}\right ) - 3 \,{\left (9 \, b^{3} x^{7} - 24 \, a b^{2} x^{4} + 5 \, a^{2} b x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{243 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 4.77737, size = 126, normalized size = 0.91 \begin{align*} \frac{a^{\frac{8}{3}} 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{2 a^{\frac{5}{3}} b 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 )} + \frac{a^{\frac{2}{3}} b^{2} x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{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 )} \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 (b x^{3} - a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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