Optimal. Leaf size=93 \[ -\log \left (\sqrt [3]{x^5-x^2}+x\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^5-x^2} x+\left (x^5-x^2\right )^{2/3}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^5-x^2}-x}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 1.01, 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 {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-1+x^3}\right ) \int \frac {1+2 x^3}{x^{2/3} \sqrt [3]{-1+x^3} \left (-1+x+x^3\right )} \, dx}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x^9}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {2}{\sqrt [3]{-1+x^9}}+\frac {3-2 x^3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {3-2 x^3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}+\frac {\left (6 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^9}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {\left (6 x^{2/3} \sqrt [3]{1-x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x^9}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}+\frac {\left (3 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )}-\frac {2 x^3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ &=\frac {6 x \sqrt [3]{1-x^3} \, _2F_1\left (\frac {1}{9},\frac {1}{3};\frac {10}{9};x^3\right )}{\sqrt [3]{-x^2+x^5}}-\frac {\left (6 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}+\frac {\left (9 x^{2/3} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^9} \left (-1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x^2+x^5}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+2 x^3}{\left (-1+x+x^3\right ) \sqrt [3]{-x^2+x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.93, size = 93, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-x^2+x^5}}\right )-\log \left (x+\sqrt [3]{-x^2+x^5}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x^2+x^5}+\left (-x^2+x^5\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.62, size = 124, normalized size = 1.33 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{4} + x^{2} - x\right )} + 2 \, \sqrt {3} {\left (x^{5} - x^{2}\right )}^{\frac {2}{3}}}{3 \, {\left (x^{4} - x^{2} - x\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {x^{4} + x^{2} + 3 \, {\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} x - x + 3 \, {\left (x^{5} - x^{2}\right )}^{\frac {2}{3}}}{x^{4} + x^{2} - x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{3} + 1}{{\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 4.32, size = 582, normalized size = 6.26
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {3175724374 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}-28078997418 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-11115035309 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+40942487600 x^{4}-3175724374 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x +35847343722 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {2}{3}}-35847343722 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {1}{3}} x +36349865670 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+28078997418 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -13024082868 \left (x^{5}-x^{2}\right )^{\frac {2}{3}}+13024082868 x \left (x^{5}-x^{2}\right )^{\frac {1}{3}}-29244634000 x^{2}-40942487600 x}{x \left (x^{3}+x -1\right )}\right )}{2}-\frac {\ln \left (-\frac {3175724374 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}+15376099922 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-11115035309 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-2512609740 x^{4}-3175724374 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -35847343722 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {2}{3}}+35847343722 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {1}{3}} x +8110275566 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-15376099922 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +58670604576 \left (x^{5}-x^{2}\right )^{\frac {2}{3}}-58670604576 x \left (x^{5}-x^{2}\right )^{\frac {1}{3}}-1005043896 x^{2}+2512609740 x}{x \left (x^{3}+x -1\right )}\right ) \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )}{2}+\ln \left (-\frac {3175724374 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}+15376099922 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-11115035309 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-2512609740 x^{4}-3175724374 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -35847343722 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {2}{3}}+35847343722 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{5}-x^{2}\right )^{\frac {1}{3}} x +8110275566 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-15376099922 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +58670604576 \left (x^{5}-x^{2}\right )^{\frac {2}{3}}-58670604576 x \left (x^{5}-x^{2}\right )^{\frac {1}{3}}-1005043896 x^{2}+2512609740 x}{x \left (x^{3}+x -1\right )}\right )\) | \(582\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{3} + 1}{{\left (x^{5} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {2\,x^3+1}{{\left (x^5-x^2\right )}^{1/3}\,\left (x^3+x-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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.
[In]
[Out]
________________________________________________________________________________________