Optimal. Leaf size=140 \[ \frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^5+1}-x\right )}{2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^5+1}+x}\right )}{2^{2/3}}+\frac {3 \left (x^5+1\right )^{2/3} \left (2 x^5+5 x^3+2\right )}{10 x^5}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^5+1} x+2^{2/3} \left (x^5+1\right )^{2/3}+x^2\right )}{2\ 2^{2/3}} \]
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Rubi [F] time = 1.29, 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 (2-x^3+2 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 (2-x^3+2 x^5\right )} \, dx &=\int \left (-\frac {3 \left (1+x^5\right )^{2/3}}{x^6}-\frac {3 \left (1+x^5\right )^{2/3}}{x^3}+\frac {2 \left (1+x^5\right )^{2/3}}{x}+\frac {\left (-3+10 x^2\right ) \left (1+x^5\right )^{2/3}}{2-x^3+2 x^5}\right ) \, dx\\ &=2 \int \frac {\left (1+x^5\right )^{2/3}}{x} \, dx-3 \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx-3 \int \frac {\left (1+x^5\right )^{2/3}}{x^3} \, dx+\int \frac {\left (-3+10 x^2\right ) \left (1+x^5\right )^{2/3}}{2-x^3+2 x^5} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {2}{5};\frac {3}{5};-x^5\right )}{2 x^2}+\frac {2}{5} \operatorname {Subst}\left (\int \frac {(1+x)^{2/3}}{x} \, dx,x,x^5\right )-\frac {3}{5} \operatorname {Subst}\left (\int \frac {(1+x)^{2/3}}{x^2} \, dx,x,x^5\right )+\int \left (-\frac {3 \left (1+x^5\right )^{2/3}}{2-x^3+2 x^5}+\frac {10 x^2 \left (1+x^5\right )^{2/3}}{2-x^3+2 x^5}\right ) \, dx\\ &=\frac {3}{5} \left (1+x^5\right )^{2/3}+\frac {3 \left (1+x^5\right )^{2/3}}{5 x^5}+\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {2}{5};\frac {3}{5};-x^5\right )}{2 x^2}-3 \int \frac {\left (1+x^5\right )^{2/3}}{2-x^3+2 x^5} \, dx+10 \int \frac {x^2 \left (1+x^5\right )^{2/3}}{2-x^3+2 x^5} \, dx\\ \end {align*}
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Mathematica [F] time = 0.33, 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 (2-x^3+2 x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.36, size = 140, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^5\right )^{2/3} \left (2+5 x^3+2 x^5\right )}{10 x^5}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^5}}\right )}{2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^5}\right )}{2^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^5}+2^{2/3} \left (1+x^5\right )^{2/3}\right )}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 (2 \, x^{5} - x^{3} + 2\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 279.84, size = 725, normalized size = 5.18
method | result | size |
risch | \(\frac {\frac {3}{5} x^{10}+\frac {3}{2} x^{8}+\frac {6}{5} x^{5}+\frac {3}{2} x^{3}+\frac {3}{5}}{x^{5} \left (x^{5}+1\right )^{\frac {1}{3}}}+\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{5}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{5}+3 \left (x^{5}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}+6 \left (x^{5}+1\right )^{\frac {2}{3}} x +2 \RootOf \left (\textit {\_Z}^{3}-2\right )-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{2 x^{5}-x^{3}+2}\right )-\frac {\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-4 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{5}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{5}-3 \left (x^{5}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{2}-6 \left (x^{5}+1\right )^{\frac {2}{3}} x -4 \RootOf \left (\textit {\_Z}^{3}-2\right )-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{2 x^{5}-x^{3}+2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )}{2}-\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-4 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{5}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{5}-3 \left (x^{5}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{2}-6 \left (x^{5}+1\right )^{\frac {2}{3}} x -4 \RootOf \left (\textit {\_Z}^{3}-2\right )-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{2 x^{5}-x^{3}+2}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )\) | \(725\) |
trager | \(\text {Expression too large to display}\) | \(1470\) |
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 (2 \, x^{5} - x^{3} + 2\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 (2\,x^5-x^3+2\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 (2 x^{5} - x^{3} + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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