Optimal. Leaf size=85 \[ \frac{2 x}{\sqrt [3]{a+b x^3}}+\frac{\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}} \]
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Rubi [A] time = 0.0125869, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {385, 239} \[ \frac{2 x}{\sqrt [3]{a+b x^3}}+\frac{\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}-\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Rule 385
Rule 239
Rubi steps
\begin{align*} \int \frac{a-b x^3}{\left (a+b x^3\right )^{4/3}} \, dx &=\frac{2 x}{\sqrt [3]{a+b x^3}}-\int \frac{1}{\sqrt [3]{a+b x^3}} \, dx\\ &=\frac{2 x}{\sqrt [3]{a+b x^3}}-\frac{\tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{b}}+\frac{\log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{b}}\\ \end{align*}
Mathematica [C] time = 0.0364964, size = 62, normalized size = 0.73 \[ \frac{4 a x-b x^4 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{4}{3},\frac{4}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 a \sqrt [3]{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.226, size = 0, normalized size = 0. \begin{align*} \int{(-b{x}^{3}+a) \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 = 1.82697, size = 976, normalized size = 11.48 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}}{\left (b^{2} x^{3} + a 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 ) + 12 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b x + 2 \,{\left (b x^{3} + a\right )} b^{\frac{2}{3}} \log \left (-\frac{b^{\frac{1}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) -{\left (b x^{3} + a\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 )}{6 \,{\left (b^{2} x^{3} + a b\right )}}, \frac{12 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} b x + 2 \,{\left (b x^{3} + a\right )} b^{\frac{2}{3}} \log \left (-\frac{b^{\frac{1}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x}\right ) -{\left (b x^{3} + a\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{6 \, \sqrt{\frac{1}{3}}{\left (b^{2} x^{3} + a 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}}}}{6 \,{\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 [C] time = 12.763, size = 70, normalized size = 0.82 \begin{align*} \frac{x \Gamma \left (\frac{1}{3}\right )}{3 \sqrt [3]{a} \sqrt [3]{1 + \frac{b x^{3}}{a}} \Gamma \left (\frac{4}{3}\right )} - \frac{b x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{4}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{4}{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 -\frac{b x^{3} - a}{{\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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