Optimal. Leaf size=71 \[ -\frac {(3 x-1)^{4/3}}{x}+12 \sqrt [3]{3 x-1}+2 \log (x)-6 \log \left (\sqrt [3]{3 x-1}+1\right )+4 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{3 x-1}}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {47, 50, 58, 618, 204, 31} \[ -\frac {(3 x-1)^{4/3}}{x}+12 \sqrt [3]{3 x-1}+2 \log (x)-6 \log \left (\sqrt [3]{3 x-1}+1\right )+4 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{3 x-1}}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 50
Rule 58
Rule 204
Rule 618
Rubi steps
\begin {align*} \int \frac {(-1+3 x)^{4/3}}{x^2} \, dx &=-\frac {(-1+3 x)^{4/3}}{x}+4 \int \frac {\sqrt [3]{-1+3 x}}{x} \, dx\\ &=12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}-4 \int \frac {1}{x (-1+3 x)^{2/3}} \, dx\\ &=12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}+2 \log (x)-6 \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+3 x}\right )-6 \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+3 x}\right )\\ &=12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}+2 \log (x)-6 \log \left (1+\sqrt [3]{-1+3 x}\right )+12 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+3 x}\right )\\ &=12 \sqrt [3]{-1+3 x}-\frac {(-1+3 x)^{4/3}}{x}+4 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+3 x}}{\sqrt {3}}\right )+2 \log (x)-6 \log \left (1+\sqrt [3]{-1+3 x}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.37 \[ \frac {9}{7} (3 x-1)^{7/3} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};1-3 x\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 88, normalized size = 1.24 \[ \frac {\sqrt [3]{3 x-1} (9 x+1)}{x}-4 \log \left (\sqrt [3]{3 x-1}+1\right )+2 \log \left ((3 x-1)^{2/3}-\sqrt [3]{3 x-1}+1\right )+4 \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{3 x-1}}{\sqrt {3}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 80, normalized size = 1.13 \[ -\frac {4 \, \sqrt {3} x \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (3 \, x - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 2 \, x \log \left ({\left (3 \, x - 1\right )}^{\frac {2}{3}} - {\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) + 4 \, x \log \left ({\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) - {\left (9 \, x + 1\right )} {\left (3 \, x - 1\right )}^{\frac {1}{3}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 76, normalized size = 1.07 \[ -4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (3 \, x - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + 9 \, {\left (3 \, x - 1\right )}^{\frac {1}{3}} + \frac {{\left (3 \, x - 1\right )}^{\frac {1}{3}}}{x} + 2 \, \log \left ({\left (3 \, x - 1\right )}^{\frac {2}{3}} - {\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) - 4 \, \log \left ({\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.60, size = 67, normalized size = 0.94
method | result | size |
meijerg | \(-\frac {4 \mathrm {signum}\left (-\frac {1}{3}+x \right )^{\frac {4}{3}} \left (-\frac {3 \Gamma \left (\frac {2}{3}\right ) x \hypergeom \left (\left [\frac {2}{3}, 1, 1\right ], \left [2, 3\right ], 3 x \right )}{2}+3 \left (2+\frac {\pi \sqrt {3}}{6}-\frac {\ln \relax (3)}{2}+\ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+\frac {3 \Gamma \left (\frac {2}{3}\right )}{4 x}\right )}{3 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (-\frac {1}{3}+x \right )\right )^{\frac {4}{3}}}\) | \(67\) |
derivativedivides | \(9 \left (-1+3 x \right )^{\frac {1}{3}}-\frac {1}{1+\left (-1+3 x \right )^{\frac {1}{3}}}-4 \ln \left (1+\left (-1+3 x \right )^{\frac {1}{3}}\right )+\frac {1+\left (-1+3 x \right )^{\frac {1}{3}}}{\left (-1+3 x \right )^{\frac {2}{3}}-\left (-1+3 x \right )^{\frac {1}{3}}+1}+2 \ln \left (\left (-1+3 x \right )^{\frac {2}{3}}-\left (-1+3 x \right )^{\frac {1}{3}}+1\right )-4 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (-1+3 x \right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )\) | \(109\) |
default | \(9 \left (-1+3 x \right )^{\frac {1}{3}}-\frac {1}{1+\left (-1+3 x \right )^{\frac {1}{3}}}-4 \ln \left (1+\left (-1+3 x \right )^{\frac {1}{3}}\right )+\frac {1+\left (-1+3 x \right )^{\frac {1}{3}}}{\left (-1+3 x \right )^{\frac {2}{3}}-\left (-1+3 x \right )^{\frac {1}{3}}+1}+2 \ln \left (\left (-1+3 x \right )^{\frac {2}{3}}-\left (-1+3 x \right )^{\frac {1}{3}}+1\right )-4 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (-1+3 x \right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )\) | \(109\) |
risch | \(\frac {\left (-1+3 x \right )^{\frac {1}{3}}}{x}+\frac {\left (-\frac {4 \left (-1+3 x \right )^{\frac {2}{3}} \left (-\mathrm {signum}\left (-\frac {1}{3}+x \right )\right )^{\frac {2}{3}} \left (2 \Gamma \left (\frac {2}{3}\right ) x \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], 3 x \right )+\left (\frac {\pi \sqrt {3}}{6}-\frac {\ln \relax (3)}{2}+\ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{\left (\left (-1+3 x \right )^{2}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (-\frac {1}{3}+x \right )^{\frac {2}{3}}}+\frac {9 \left (-1+3 x \right )^{\frac {2}{3}} \left (-\mathrm {signum}\left (-\frac {1}{3}+x \right )\right )^{\frac {2}{3}} x \hypergeom \left (\left [\frac {2}{3}, 1\right ], \relax [2], 3 x \right )}{\left (\left (-1+3 x \right )^{2}\right )^{\frac {1}{3}} \mathrm {signum}\left (-\frac {1}{3}+x \right )^{\frac {2}{3}}}\right ) \left (\left (-1+3 x \right )^{2}\right )^{\frac {1}{3}}}{\left (-1+3 x \right )^{\frac {2}{3}}}\) | \(146\) |
trager | \(\frac {\left (1+9 x \right ) \left (-1+3 x \right )^{\frac {1}{3}}}{x}-4 \ln \left (\frac {\left (-1+3 x \right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +\left (-1+3 x \right )^{\frac {1}{3}}}{x}\right )+4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {\left (-1+3 x \right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -\left (-1+3 x \right )^{\frac {2}{3}}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-1+3 x \right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{x}\right )\) | \(195\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 76, normalized size = 1.07 \[ -4 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (3 \, x - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + 9 \, {\left (3 \, x - 1\right )}^{\frac {1}{3}} + \frac {{\left (3 \, x - 1\right )}^{\frac {1}{3}}}{x} + 2 \, \log \left ({\left (3 \, x - 1\right )}^{\frac {2}{3}} - {\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) - 4 \, \log \left ({\left (3 \, x - 1\right )}^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 90, normalized size = 1.27 \[ 9\,{\left (3\,x-1\right )}^{1/3}-4\,\ln \left (144\,{\left (3\,x-1\right )}^{1/3}+144\right )+\frac {{\left (3\,x-1\right )}^{1/3}}{x}+\ln \left (18-36\,{\left (3\,x-1\right )}^{1/3}+\sqrt {3}\,18{}\mathrm {i}\right )\,\left (2+\sqrt {3}\,2{}\mathrm {i}\right )-\ln \left (36\,{\left (3\,x-1\right )}^{1/3}-18+\sqrt {3}\,18{}\mathrm {i}\right )\,\left (-2+\sqrt {3}\,2{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.27, size = 541, normalized size = 7.62 \[ \frac {189 \sqrt [3]{3} \left (x - \frac {1}{3}\right )^{\frac {4}{3}} e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {84 \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {84 \left (x - \frac {1}{3}\right ) \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {i \pi }{3}} + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} - \frac {84 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{i \pi } + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {84 \left (x - \frac {1}{3}\right ) e^{\frac {2 i \pi }{3}} \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {28 \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {i \pi }{3}} + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} - \frac {28 e^{\frac {i \pi }{3}} \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{i \pi } + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} + \frac {28 e^{\frac {2 i \pi }{3}} \log {\left (- \sqrt [3]{3} \sqrt [3]{x - \frac {1}{3}} e^{\frac {5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac {7}{3}\right )}{9 \left (x - \frac {1}{3}\right ) e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) + 3 e^{\frac {i \pi }{3}} \Gamma \left (\frac {10}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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