Optimal. Leaf size=93 \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{4/3}}-\frac{\log (x)}{2 a^{4/3}}+\frac{3}{a \sqrt [3]{a+b x}} \]
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Rubi [A] time = 0.0331465, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {51, 55, 617, 204, 31} \[ \frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{4/3}}-\frac{\log (x)}{2 a^{4/3}}+\frac{3}{a \sqrt [3]{a+b x}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x (a+b x)^{4/3}} \, dx &=\frac{3}{a \sqrt [3]{a+b x}}+\frac{\int \frac{1}{x \sqrt [3]{a+b x}} \, dx}{a}\\ &=\frac{3}{a \sqrt [3]{a+b x}}-\frac{\log (x)}{2 a^{4/3}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{2 a^{4/3}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{2 a}\\ &=\frac{3}{a \sqrt [3]{a+b x}}-\frac{\log (x)}{2 a^{4/3}}+\frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{a^{4/3}}\\ &=\frac{3}{a \sqrt [3]{a+b x}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{4/3}}-\frac{\log (x)}{2 a^{4/3}}+\frac{3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.0222593, size = 30, normalized size = 0.32 \[ \frac{3 \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};\frac{b x}{a}+1\right )}{a \sqrt [3]{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 87, normalized size = 0.9 \begin{align*}{\ln \left ( \sqrt [3]{bx+a}-\sqrt [3]{a} \right ){a}^{-{\frac{4}{3}}}}-{\frac{1}{2}\ln \left ( \left ( bx+a \right ) ^{{\frac{2}{3}}}+\sqrt [3]{a}\sqrt [3]{bx+a}+{a}^{{\frac{2}{3}}} \right ){a}^{-{\frac{4}{3}}}}+{\sqrt{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{bx+a}}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{4}{3}}}}+3\,{\frac{1}{a\sqrt [3]{bx+a}}} \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 = 1.63811, size = 834, normalized size = 8.97 \begin{align*} \left [\frac{\sqrt{3}{\left (a b x + a^{2}\right )} \sqrt{-\frac{1}{a^{\frac{2}{3}}}} \log \left (\frac{2 \, b x + \sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{2}{3}} a^{\frac{2}{3}} -{\left (b x + a\right )}^{\frac{1}{3}} a - a^{\frac{4}{3}}\right )} \sqrt{-\frac{1}{a^{\frac{2}{3}}}} - 3 \,{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + 3 \, a}{x}\right ) -{\left (b x + a\right )} a^{\frac{2}{3}} \log \left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) + 2 \,{\left (b x + a\right )} a^{\frac{2}{3}} \log \left ({\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) + 6 \,{\left (b x + a\right )}^{\frac{2}{3}} a}{2 \,{\left (a^{2} b x + a^{3}\right )}}, -\frac{{\left (b x + a\right )} a^{\frac{2}{3}} \log \left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) - 2 \,{\left (b x + a\right )} a^{\frac{2}{3}} \log \left ({\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) - \frac{2 \, \sqrt{3}{\left (a b x + a^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{1}{3}}} - 6 \,{\left (b x + a\right )}^{\frac{2}{3}} a}{2 \,{\left (a^{2} b x + a^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.92168, size = 184, normalized size = 1.98 \begin{align*} - \frac{\Gamma \left (- \frac{1}{3}\right )}{a \sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} \Gamma \left (\frac{2}{3}\right )} - \frac{\log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right )} - \frac{e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right )} - \frac{e^{- \frac{2 i \pi }{3}} \log{\left (1 - \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b} + x} e^{\frac{4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.31149, size = 120, normalized size = 1.29 \begin{align*} \frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{4}{3}}} - \frac{\log \left ({\left (b x + a\right )}^{\frac{2}{3}} +{\left (b x + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{2 \, a^{\frac{4}{3}}} + \frac{\log \left ({\left |{\left (b x + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{4}{3}}} + \frac{3}{{\left (b x + a\right )}^{\frac{1}{3}} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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