Optimal. Leaf size=76 \[ -\frac{3 \log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{2 a}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{a-2 \sqrt [3]{-a^3-b^3 x}}{\sqrt{3} a}\right )}{a}+\frac{\log (x)}{2 a} \]
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Rubi [A] time = 0.0283408, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {56, 617, 204, 31} \[ -\frac{3 \log \left (\sqrt [3]{-a^3-b^3 x}+a\right )}{2 a}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{a-2 \sqrt [3]{-a^3-b^3 x}}{\sqrt{3} a}\right )}{a}+\frac{\log (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 56
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt [3]{-a^3-b^3 x}} \, dx &=\frac{\log (x)}{2 a}+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{a^2-a x+x^2} \, dx,x,\sqrt [3]{-a^3-b^3 x}\right )-\frac{3 \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,\sqrt [3]{-a^3-b^3 x}\right )}{2 a}\\ &=\frac{\log (x)}{2 a}-\frac{3 \log \left (a+\sqrt [3]{-a^3-b^3 x}\right )}{2 a}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{-a^3-b^3 x}}{a}\right )}{a}\\ &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{-a^3-b^3 x}}{a}}{\sqrt{3}}\right )}{a}+\frac{\log (x)}{2 a}-\frac{3 \log \left (a+\sqrt [3]{-a^3-b^3 x}\right )}{2 a}\\ \end{align*}
Mathematica [C] time = 0.0531762, size = 41, normalized size = 0.54 \[ \frac{3 \left (-a^3-b^3 x\right )^{2/3} \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{x b^3}{a^3}+1\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 101, normalized size = 1.3 \begin{align*} -{\frac{1}{a}\ln \left ( a+\sqrt [3]{-{b}^{3}x-{a}^{3}} \right ) }+{\frac{1}{2\,a}\ln \left ( \left ( -{b}^{3}x-{a}^{3} \right ) ^{{\frac{2}{3}}}-a\sqrt [3]{-{b}^{3}x-{a}^{3}}+{a}^{2} \right ) }+{\frac{\sqrt{3}}{a}\arctan \left ({\frac{\sqrt{3}}{3\,a} \left ( 2\,\sqrt [3]{-{b}^{3}x-{a}^{3}}-a \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54618, size = 132, normalized size = 1.74 \begin{align*} \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \,{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a} + \frac{\log \left (a^{2} -{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x - a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a} - \frac{\log \left (a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65885, size = 232, normalized size = 3.05 \begin{align*} \frac{2 \, \sqrt{3} \arctan \left (-\frac{\sqrt{3} a - 2 \, \sqrt{3}{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}}{3 \, a}\right ) + \log \left (a^{2} -{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x - a^{3}\right )}^{\frac{2}{3}}\right ) - 2 \, \log \left (a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.37006, size = 139, normalized size = 1.83 \begin{align*} \frac{\log{\left (- \frac{a e^{\frac{2 i \pi }{3}}}{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a \Gamma \left (\frac{2}{3}\right )} - \frac{e^{\frac{i \pi }{3}} \log{\left (- \frac{a e^{\frac{4 i \pi }{3}}}{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a \Gamma \left (\frac{2}{3}\right )} + \frac{e^{\frac{2 i \pi }{3}} \log{\left (- \frac{a e^{2 i \pi }}{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac{1}{3}\right )}{3 a \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 = 1.26894, size = 134, normalized size = 1.76 \begin{align*} \frac{\sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (a - 2 \,{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a} + \frac{\log \left (a^{2} -{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} a +{\left (-b^{3} x - a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a} - \frac{\log \left ({\left | a +{\left (-b^{3} x - a^{3}\right )}^{\frac{1}{3}} \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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