Optimal. Leaf size=105 \[ \frac {3 \left (-1+x^5\right )^{2/3} \left (-4-5 x^3+4 x^5\right )}{10 x^5}-\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^5}}\right )-\log \left (x+\sqrt [3]{-1+x^5}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^5}+\left (-1+x^5\right )^{2/3}\right ) \]
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Rubi [F]
time = 0.75, 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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Rubi steps
\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} \text {Subst}\left (\int \frac {(-1+x)^{2/3}}{x} \, dx,x,x^5\right )+\frac {6}{5} \text {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 [A]
time = 2.07, size = 102, normalized size = 0.97 \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} \text {ArcTan}\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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 20.01, size = 333, normalized size = 3.17
method | result | size |
risch | \(\frac {-\frac {3}{2} x^{8}+\frac {3}{2} x^{3}-\frac {12}{5} x^{5}+\frac {6}{5}+\frac {6}{5} x^{10}}{x^{5} \left (x^{5}-1\right )^{\frac {1}{3}}}-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-2 x^{5}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}-1\right )^{\frac {2}{3}} x -3 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2}{x^{5}+x^{3}-1}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-x^{5}+3 \left (x^{5}-1\right )^{\frac {2}{3}} x -3 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{x^{5}+x^{3}-1}\right )+\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-2 x^{5}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}-1\right )^{\frac {2}{3}} x -3 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2}{x^{5}+x^{3}-1}\right )\) | \(333\) |
trager | \(\frac {3 \left (x^{5}-1\right )^{\frac {2}{3}} \left (4 x^{5}-5 x^{3}-4\right )}{10 x^{5}}-\ln \left (\frac {1973169492632104665894772909105152 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}-97346550098888531109714265679328 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-7646031783949405580342245022782464 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}+714468692923457785263401407273 x^{5}-105550256795540691231071445123936 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}} x +19277590060830940057788253380576 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}+127681310055207165244114369615680 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}+1300290071420537825925621859422 \left (x^{5}-1\right )^{\frac {2}{3}} x +1099481841620215533656994220041 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-530089675394823518098652657009 x^{3}-1973169492632104665894772909105152 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}+97346550098888531109714265679328 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )-714468692923457785263401407273}{x^{5}+x^{3}-1}\right )+96 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \ln \left (-\frac {1012432749095564771328 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{5}+22999020922536659328 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{5}-3923176902745313488896 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2} x^{3}-219255606241635155 x^{5}-29644229945602756800 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}} x -65139908061224566944 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}+38915943346923997728 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right ) x^{3}-369746647037727189 \left (x^{5}-1\right )^{\frac {2}{3}} x +308794061933362050 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-76262819562307880 x^{3}-1012432749095564771328 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )^{2}-22999020922536659328 \RootOf \left (9216 \textit {\_Z}^{2}-96 \textit {\_Z} +1\right )+219255606241635155}{x^{5}+x^{3}-1}\right )\) | \(414\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.46, 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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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}} \cdot \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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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