Optimal. Leaf size=104 \[ \frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {11 \sqrt [3]{-1+x}}{27 x^2}+\frac {55 \sqrt [3]{-1+x}}{81 x}-\frac {110 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {55}{81} \log \left (1+\sqrt [3]{-1+x}\right )-\frac {55 \log (x)}{243} \]
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Rubi [A]
time = 0.03, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {44, 60, 632,
210, 31} \begin {gather*} -\frac {110 \text {ArcTan}\left (\frac {1-2 \sqrt [3]{x-1}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {\sqrt [3]{x-1}}{4 x^4}+\frac {11 \sqrt [3]{x-1}}{36 x^3}+\frac {11 \sqrt [3]{x-1}}{27 x^2}+\frac {55 \sqrt [3]{x-1}}{81 x}+\frac {55}{81} \log \left (\sqrt [3]{x-1}+1\right )-\frac {55 \log (x)}{243} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 60
Rule 210
Rule 632
Rubi steps
\begin {align*} \int \frac {1}{(-1+x)^{2/3} x^5} \, dx &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11}{12} \int \frac {1}{(-1+x)^{2/3} x^4} \, dx\\ &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {22}{27} \int \frac {1}{(-1+x)^{2/3} x^3} \, dx\\ &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {11 \sqrt [3]{-1+x}}{27 x^2}+\frac {55}{81} \int \frac {1}{(-1+x)^{2/3} x^2} \, dx\\ &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {11 \sqrt [3]{-1+x}}{27 x^2}+\frac {55 \sqrt [3]{-1+x}}{81 x}+\frac {110}{243} \int \frac {1}{(-1+x)^{2/3} x} \, dx\\ &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {11 \sqrt [3]{-1+x}}{27 x^2}+\frac {55 \sqrt [3]{-1+x}}{81 x}-\frac {55 \log (x)}{243}+\frac {55}{81} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x}\right )+\frac {55}{81} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x}\right )\\ &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {11 \sqrt [3]{-1+x}}{27 x^2}+\frac {55 \sqrt [3]{-1+x}}{81 x}+\frac {55}{81} \log \left (1+\sqrt [3]{-1+x}\right )-\frac {55 \log (x)}{243}-\frac {110}{81} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x}\right )\\ &=\frac {\sqrt [3]{-1+x}}{4 x^4}+\frac {11 \sqrt [3]{-1+x}}{36 x^3}+\frac {11 \sqrt [3]{-1+x}}{27 x^2}+\frac {55 \sqrt [3]{-1+x}}{81 x}-\frac {110 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {55}{81} \log \left (1+\sqrt [3]{-1+x}\right )-\frac {55 \log (x)}{243}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 90, normalized size = 0.87 \begin {gather*} \frac {1}{972} \left (\frac {3 \sqrt [3]{-1+x} \left (81+99 x+132 x^2+220 x^3\right )}{x^4}-440 \sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x}}{\sqrt {3}}\right )+440 \log \left (1+\sqrt [3]{-1+x}\right )-220 \log \left (1-\sqrt [3]{-1+x}+(-1+x)^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs.
\(2(75)=150\).
time = 0.31, size = 158, normalized size = 1.52
method | result | size |
meijerg | \(\frac {\left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {2}{3}} \left (-\frac {\Gamma \left (\frac {2}{3}\right )}{4 x^{4}}-\frac {2 \Gamma \left (\frac {2}{3}\right )}{9 x^{3}}-\frac {5 \Gamma \left (\frac {2}{3}\right )}{18 x^{2}}-\frac {40 \Gamma \left (\frac {2}{3}\right )}{81 x}+\frac {110 \left (\frac {877}{1320}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+\ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{243}+\frac {308 \Gamma \left (\frac {2}{3}\right ) x \hypergeom \left (\left [1, 1, \frac {17}{3}\right ], \left [2, 6\right ], x\right )}{729}\right )}{\Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (-1+x \right )^{\frac {2}{3}}}\) | \(85\) |
risch | \(\frac {220 x^{4}-88 x^{3}-33 x^{2}-18 x -81}{324 x^{4} \left (-1+x \right )^{\frac {2}{3}}}+\frac {110 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {2}{3}} \left (\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+\ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+\frac {2 \Gamma \left (\frac {2}{3}\right ) x \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x\right )}{3}\right )}{243 \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (-1+x \right )^{\frac {2}{3}}}\) | \(87\) |
derivativedivides | \(-\frac {-75 \left (-1+x \right )^{\frac {7}{3}}+190 \left (-1+x \right )^{2}-350 \left (-1+x \right )^{\frac {5}{3}}+\frac {1157 \left (-1+x \right )^{\frac {4}{3}}}{4}+\frac {149}{4}-138 x -116 \left (-1+x \right )^{\frac {2}{3}}+137 \left (-1+x \right )^{\frac {1}{3}}}{243 \left (\left (-1+x \right )^{\frac {2}{3}}-\left (-1+x \right )^{\frac {1}{3}}+1\right )^{4}}-\frac {55 \ln \left (\left (-1+x \right )^{\frac {2}{3}}-\left (-1+x \right )^{\frac {1}{3}}+1\right )}{243}+\frac {110 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (-1+x \right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{243}-\frac {1}{324 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{4}}-\frac {5}{243 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{3}}-\frac {20}{243 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{2}}-\frac {25}{81 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}+\frac {110 \ln \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}{243}\) | \(158\) |
default | \(-\frac {-75 \left (-1+x \right )^{\frac {7}{3}}+190 \left (-1+x \right )^{2}-350 \left (-1+x \right )^{\frac {5}{3}}+\frac {1157 \left (-1+x \right )^{\frac {4}{3}}}{4}+\frac {149}{4}-138 x -116 \left (-1+x \right )^{\frac {2}{3}}+137 \left (-1+x \right )^{\frac {1}{3}}}{243 \left (\left (-1+x \right )^{\frac {2}{3}}-\left (-1+x \right )^{\frac {1}{3}}+1\right )^{4}}-\frac {55 \ln \left (\left (-1+x \right )^{\frac {2}{3}}-\left (-1+x \right )^{\frac {1}{3}}+1\right )}{243}+\frac {110 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (-1+x \right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{243}-\frac {1}{324 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{4}}-\frac {5}{243 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{3}}-\frac {20}{243 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )^{2}}-\frac {25}{81 \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}+\frac {110 \ln \left (1+\left (-1+x \right )^{\frac {1}{3}}\right )}{243}\) | \(158\) |
trager | \(\frac {\left (220 x^{3}+132 x^{2}+99 x +81\right ) \left (-1+x \right )^{\frac {1}{3}}}{324 x^{4}}-\frac {110 \ln \left (\frac {72 \left (-1+x \right )^{\frac {2}{3}} \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )-1152 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2} x -72 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) \left (-1+x \right )^{\frac {1}{3}}+2304 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2}-72 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) x +120 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )-x +1}{x}\right )}{243}-\frac {1760 \ln \left (\frac {72 \left (-1+x \right )^{\frac {2}{3}} \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )-1152 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2} x -72 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) \left (-1+x \right )^{\frac {1}{3}}+2304 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2}-72 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) x +120 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )-x +1}{x}\right ) \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )}{81}+\frac {1760 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) \ln \left (-\frac {144 \left (-1+x \right )^{\frac {2}{3}} \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )+2304 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2} x +3 \left (-1+x \right )^{\frac {2}{3}}-144 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) \left (-1+x \right )^{\frac {1}{3}}-4608 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )^{2}-48 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right ) x -3 \left (-1+x \right )^{\frac {1}{3}}+48 \RootOf \left (2304 \textit {\_Z}^{2}+48 \textit {\_Z} +1\right )+1}{x}\right )}{81}\) | \(379\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.94, size = 105, normalized size = 1.01 \begin {gather*} \frac {110}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {220 \, {\left (x - 1\right )}^{\frac {10}{3}} + 792 \, {\left (x - 1\right )}^{\frac {7}{3}} + 1023 \, {\left (x - 1\right )}^{\frac {4}{3}} + 532 \, {\left (x - 1\right )}^{\frac {1}{3}}}{324 \, {\left ({\left (x - 1\right )}^{4} + 4 \, {\left (x - 1\right )}^{3} + 6 \, {\left (x - 1\right )}^{2} + 4 \, x - 3\right )}} - \frac {55}{243} \, \log \left ({\left (x - 1\right )}^{\frac {2}{3}} - {\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {110}{243} \, \log \left ({\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 86, normalized size = 0.83 \begin {gather*} \frac {440 \, \sqrt {3} x^{4} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 220 \, x^{4} \log \left ({\left (x - 1\right )}^{\frac {2}{3}} - {\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) + 440 \, x^{4} \log \left ({\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (220 \, x^{3} + 132 \, x^{2} + 99 \, x + 81\right )} {\left (x - 1\right )}^{\frac {1}{3}}}{972 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 39.76, size = 12993, normalized size = 124.93 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 82, normalized size = 0.79 \begin {gather*} \frac {110}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {220 \, {\left (x - 1\right )}^{\frac {10}{3}} + 792 \, {\left (x - 1\right )}^{\frac {7}{3}} + 1023 \, {\left (x - 1\right )}^{\frac {4}{3}} + 532 \, {\left (x - 1\right )}^{\frac {1}{3}}}{324 \, x^{4}} - \frac {55}{243} \, \log \left ({\left (x - 1\right )}^{\frac {2}{3}} - {\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {110}{243} \, \log \left ({\left (x - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 120, normalized size = 1.15 \begin {gather*} \frac {110\,\ln \left (\frac {12100\,{\left (x-1\right )}^{1/3}}{6561}+\frac {12100}{6561}\right )}{243}+\frac {\frac {133\,{\left (x-1\right )}^{1/3}}{81}+\frac {341\,{\left (x-1\right )}^{4/3}}{108}+\frac {22\,{\left (x-1\right )}^{7/3}}{9}+\frac {55\,{\left (x-1\right )}^{10/3}}{81}}{4\,x+6\,{\left (x-1\right )}^2+4\,{\left (x-1\right )}^3+{\left (x-1\right )}^4-3}-\ln \left (\frac {55}{27}-\frac {110\,{\left (x-1\right )}^{1/3}}{27}+\frac {\sqrt {3}\,55{}\mathrm {i}}{27}\right )\,\left (\frac {55}{243}+\frac {\sqrt {3}\,55{}\mathrm {i}}{243}\right )+\ln \left (\frac {110\,{\left (x-1\right )}^{1/3}}{27}-\frac {55}{27}+\frac {\sqrt {3}\,55{}\mathrm {i}}{27}\right )\,\left (-\frac {55}{243}+\frac {\sqrt {3}\,55{}\mathrm {i}}{243}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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