Optimal. Leaf size=113 \[ \frac{7}{3} x \left (a+b x^3\right )^{2/3}+\frac{2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac{5 a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{b}}-\frac{10 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{b}} \]
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Rubi [A] time = 0.0416018, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {413, 388, 239} \[ \frac{7}{3} x \left (a+b x^3\right )^{2/3}+\frac{2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac{5 a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{b}}-\frac{10 a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Rule 413
Rule 388
Rule 239
Rubi steps
\begin{align*} \int \frac{\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx &=\frac{2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac{\int \frac{-a^2 b+7 a b^2 x^3}{\sqrt [3]{a+b x^3}} \, dx}{a b}\\ &=\frac{2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac{7}{3} x \left (a+b x^3\right )^{2/3}-\frac{1}{3} (10 a) \int \frac{1}{\sqrt [3]{a+b x^3}} \, dx\\ &=\frac{2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac{7}{3} x \left (a+b x^3\right )^{2/3}-\frac{10 a \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{3 \sqrt{3} \sqrt [3]{b}}+\frac{5 a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.083197, size = 137, normalized size = 1.21 \[ \frac{x \left (13 a+b x^3\right )}{3 \sqrt [3]{a+b x^3}}-\frac{5 a \left (\log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )\right )}{9 \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.41, 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(-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 [B] time = 2.39682, size = 1062, normalized size = 9.4 \begin{align*} \left [\frac{15 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x^{3} + a^{2} b\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 ) + 10 \,{\left (a b x^{3} + a^{2}\right )} b^{\frac{2}{3}} \log \left (-\frac{b^{\frac{1}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) - 5 \,{\left (a b x^{3} + a^{2}\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 (b^{2} x^{4} + 13 \, a b x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{9 \,{\left (b^{2} x^{3} + a b\right )}}, \frac{10 \,{\left (a b x^{3} + a^{2}\right )} b^{\frac{2}{3}} \log \left (-\frac{b^{\frac{1}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) - 5 \,{\left (a b x^{3} + a^{2}\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{30 \, \sqrt{\frac{1}{3}}{\left (a b^{2} x^{3} + a^{2} b\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 (b^{2} x^{4} + 13 \, a b x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{9 \,{\left (b^{2} x^{3} + a b\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- a + b x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac{4}{3}}}\, dx \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 \frac{{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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