Optimal. Leaf size=72 \[ -\frac {\log (x)}{2 a^2}+\frac {3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a^2}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt {3} a}\right )}{a^2} \]
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Rubi [A] time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {57, 617, 204, 31} \begin {gather*} \frac {3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a^2}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt {3} a}\right )}{a^2}-\frac {\log (x)}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 204
Rule 617
Rubi steps
\begin {align*} \int \frac {1}{x \left (a^3+b^3 x\right )^{2/3}} \, dx &=-\frac {\log (x)}{2 a^2}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{a-x} \, dx,x,\sqrt [3]{a^3+b^3 x}\right )}{2 a^2}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{a^2+a x+x^2} \, dx,x,\sqrt [3]{a^3+b^3 x}\right )}{2 a}\\ &=-\frac {\log (x)}{2 a^2}+\frac {3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a^2}+\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^2}\\ &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a^3+b^3 x}}{a}}{\sqrt {3}}\right )}{a^2}-\frac {\log (x)}{2 a^2}+\frac {3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 95, normalized size = 1.32 \begin {gather*} -\frac {-2 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt {3} a}\right )+\log \left (a \sqrt [3]{a^3+b^3 x}+\left (a^3+b^3 x\right )^{2/3}+a^2\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 103, normalized size = 1.43 \begin {gather*} \frac {\log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{a^2}-\frac {\log \left (a \sqrt [3]{a^3+b^3 x}+\left (a^3+b^3 x\right )^{2/3}+a^2\right )}{2 a^2}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3+b^3 x}}{\sqrt {3} a}+\frac {1}{\sqrt {3}}\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 86, normalized size = 1.19 \begin {gather*} -\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^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 88, normalized size = 1.22 \begin {gather*} -\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^{2}} - \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^{2}} + \frac {\log \left ({\left | -a + {\left (b^{3} x + a^{3}\right )}^{\frac {1}{3}} \right |}\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 88, normalized size = 1.22 \begin {gather*} -\frac {\sqrt {3}\, \arctan \left (\frac {\left (a +2 \left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a^{2}}+\frac {\ln \left (-a +\left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right )}{a^{2}}-\frac {\ln \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^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.88, size = 87, normalized size = 1.21 \begin {gather*} -\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^{2}} - \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^{2}} + \frac {\log \left (-a + {\left (b^{3} x + a^{3}\right )}^{\frac {1}{3}}\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 101, normalized size = 1.40 \begin {gather*} \frac {\ln \left (9\,a-9\,{\left (a^3+x\,b^3\right )}^{1/3}\right )}{a^2}+\frac {\ln \left (9\,{\left (a^3+x\,b^3\right )}^{1/3}-\frac {9\,a\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^2}-\frac {\ln \left (9\,{\left (a^3+x\,b^3\right )}^{1/3}+\frac {9\,a\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.86, size = 134, normalized size = 1.86 \begin {gather*} \frac {\log {\left (1 - \frac {b \sqrt [3]{\frac {a^{3}}{b^{3}} + x}}{a} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{2} \Gamma \left (\frac {4}{3}\right )} + \frac {e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {b \sqrt [3]{\frac {a^{3}}{b^{3}} + x} e^{\frac {2 i \pi }{3}}}{a} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{2} \Gamma \left (\frac {4}{3}\right )} + \frac {e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {b \sqrt [3]{\frac {a^{3}}{b^{3}} + x} e^{\frac {4 i \pi }{3}}}{a} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{2} \Gamma \left (\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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