Optimal. Leaf size=105 \[ -\log \left (\sqrt [3]{x^5-1}+x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^5-1}-x}\right )+\frac {3 \left (x^5-1\right )^{2/3} \left (4 x^5-5 x^3-4\right )}{10 x^5}+\frac {1}{2} \log \left (-\sqrt [3]{x^5-1} x+\left (x^5-1\right )^{2/3}+x^2\right ) \]
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Rubi [F] time = 1.23, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx &=\int \left (\frac {6 \left (-1+x^5\right )^{2/3}}{x^6}+\frac {3 \left (-1+x^5\right )^{2/3}}{x^3}+\frac {4 \left (-1+x^5\right )^{2/3}}{x}+\frac {\left (-3-5 x^2\right ) \left (-1+x^5\right )^{2/3}}{-1+x^3+x^5}\right ) \, dx\\ &=3 \int \frac {\left (-1+x^5\right )^{2/3}}{x^3} \, dx+4 \int \frac {\left (-1+x^5\right )^{2/3}}{x} \, dx+6 \int \frac {\left (-1+x^5\right )^{2/3}}{x^6} \, dx+\int \frac {\left (-3-5 x^2\right ) \left (-1+x^5\right )^{2/3}}{-1+x^3+x^5} \, dx\\ &=\frac {4}{5} \operatorname {Subst}\left (\int \frac {(-1+x)^{2/3}}{x} \, dx,x,x^5\right )+\frac {6}{5} \operatorname {Subst}\left (\int \frac {(-1+x)^{2/3}}{x^2} \, dx,x,x^5\right )+\frac {\left (3 \left (-1+x^5\right )^{2/3}\right ) \int \frac {\left (1-x^5\right )^{2/3}}{x^3} \, dx}{\left (1-x^5\right )^{2/3}}+\int \left (-\frac {3 \left (-1+x^5\right )^{2/3}}{-1+x^3+x^5}-\frac {5 x^2 \left (-1+x^5\right )^{2/3}}{-1+x^3+x^5}\right ) \, dx\\ &=\frac {6}{5} \left (-1+x^5\right )^{2/3}-\frac {6 \left (-1+x^5\right )^{2/3}}{5 x^5}-\frac {3 \left (-1+x^5\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {2}{5};\frac {3}{5};x^5\right )}{2 x^2 \left (1-x^5\right )^{2/3}}-3 \int \frac {\left (-1+x^5\right )^{2/3}}{-1+x^3+x^5} \, dx-5 \int \frac {x^2 \left (-1+x^5\right )^{2/3}}{-1+x^3+x^5} \, dx\\ \end {align*}
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Mathematica [F] time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.38, size = 105, normalized size = 1.00 \begin {gather*} \frac {3 \left (-1+x^5\right )^{2/3} \left (-4-5 x^3+4 x^5\right )}{10 x^5}-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^5}}\right )-\log \left (x+\sqrt [3]{-1+x^5}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^5}+\left (-1+x^5\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 7.46, size = 144, normalized size = 1.37 \begin {gather*} -\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {1092 \, \sqrt {3} {\left (x^{5} - 1\right )}^{\frac {1}{3}} x^{2} + 2002 \, \sqrt {3} {\left (x^{5} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (121 \, x^{5} + 576 \, x^{3} - 121\right )}}{3 \, {\left (1331 \, x^{5} - 216 \, x^{3} - 1331\right )}}\right ) + 5 \, x^{5} \log \left (\frac {x^{5} + x^{3} + 3 \, {\left (x^{5} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{5} - 1\right )}^{\frac {2}{3}} x - 1}{x^{5} + x^{3} - 1}\right ) - 3 \, {\left (4 \, x^{5} - 5 \, x^{3} - 4\right )} {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{10 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{5} + x^{3} - 2\right )} {\left (2 \, x^{5} + 3\right )} {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{{\left (x^{5} + x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 14.27, size = 333, normalized size = 3.17
method | result | size |
risch | \(\frac {-\frac {12}{5} x^{5}+\frac {6}{5}-\frac {3}{2} x^{8}+\frac {3}{2} x^{3}+\frac {6}{5} x^{10}}{x^{5} \left (x^{5}-1\right )^{\frac {1}{3}}}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-x^{5}+3 \left (x^{5}-1\right )^{\frac {2}{3}} x -3 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{x^{5}+x^{3}-1}\right )-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-2 x^{5}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}-1\right )^{\frac {2}{3}} x -3 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2}{x^{5}+x^{3}-1}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-2 x^{5}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}-1\right )^{\frac {2}{3}} x -3 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2}{x^{5}+x^{3}-1}\right )\) | \(333\) |
trager | \(\frac {3 \left (x^{5}-1\right )^{\frac {2}{3}} \left (4 x^{5}-5 x^{3}-4\right )}{10 x^{5}}-96 \ln \left (\frac {309594604009335349248 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}-1274461068424225440 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-1199679090536174478336 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}-315835954172064748 x^{5}+29644229945602756800 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}} x -29644229945602756800 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}+44091369862027592064 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}+369746647037727189 \left (x^{5}-1\right )^{\frac {2}{3}} x -369746647037727189 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-109855984059848608 x^{3}-309594604009335349248 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}+1274461068424225440 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+315835954172064748}{x^{5}+x^{3}-1}\right ) \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+96 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \ln \left (\frac {309594604009335349248 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}-5175426515103594336 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-1199679090536174478336 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}-295518425803943035 x^{5}-29644229945602756800 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}} x +29644229945602756800 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-19098055475857290432 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}+678540708971089239 \left (x^{5}-1\right )^{\frac {2}{3}} x -678540708971089239 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}+219255606241635155 x^{3}-309594604009335349248 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}+5175426515103594336 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+295518425803943035}{x^{5}+x^{3}-1}\right )+\ln \left (\frac {309594604009335349248 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}-1274461068424225440 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-1199679090536174478336 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}-315835954172064748 x^{5}+29644229945602756800 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}} x -29644229945602756800 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}+44091369862027592064 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}+369746647037727189 \left (x^{5}-1\right )^{\frac {2}{3}} x -369746647037727189 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-109855984059848608 x^{3}-309594604009335349248 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}+1274461068424225440 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+315835954172064748}{x^{5}+x^{3}-1}\right )\) | \(610\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{5} + x^{3} - 2\right )} {\left (2 \, x^{5} + 3\right )} {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{{\left (x^{5} + x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^5-1\right )}^{2/3}\,\left (2\,x^5+3\right )\,\left (2\,x^5+x^3-2\right )}{x^6\,\left (x^5+x^3-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (2 x^{5} + 3\right ) \left (2 x^{5} + x^{3} - 2\right )}{x^{6} \left (x^{5} + x^{3} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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