Optimal. Leaf size=112 \[ \frac {3 \sqrt [3]{x^8+2 x^3-1}}{x}+\log \left (\sqrt [3]{x^8+2 x^3-1}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^8+2 x^3-1}+x}\right )-\frac {1}{2} \log \left (x^2+\sqrt [3]{x^8+2 x^3-1} x+\left (x^8+2 x^3-1\right )^{2/3}\right ) \]
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Rubi [F] time = 0.99, 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 {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^3+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^3+x^8\right )} \, dx &=\int \left (-\frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x^2}+\frac {x \left (3+8 x^5\right ) \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8}\right ) \, dx\\ &=-\left (3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx\right )+\int \frac {x \left (3+8 x^5\right ) \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8} \, dx\\ &=-\left (3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx\right )+\int \left (\frac {3 x \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8}+\frac {8 x^6 \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8}\right ) \, dx\\ &=-\left (3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx\right )+3 \int \frac {x \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8} \, dx+8 \int \frac {x^6 \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8} \, dx\\ \end {align*}
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Mathematica [F] time = 0.37, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^3+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.73, size = 112, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+2 x^3+x^8}}\right )+\log \left (-x+\sqrt [3]{-1+2 x^3+x^8}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+2 x^3+x^8}+\left (-1+2 x^3+x^8\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 17.48, size = 152, normalized size = 1.36 \begin {gather*} \frac {2 \, \sqrt {3} x \arctan \left (\frac {23155756059884469826063290091369873601204942180224 \, \sqrt {3} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 61059012875773331838678659685174425801373874951458 \, \sqrt {3} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (35248398304721470575821713544519821387080907584081 \, x^{8} + 77355782772550371408192688432791971088370316149922 \, x^{3} - 35248398304721470575821713544519821387080907584081\right )}}{3 \, {\left (20044909029062956675424368815298850195325332161233 \, x^{8} + 38996537437007387681732053612201126295409798546850 \, x^{3} - 20044909029062956675424368815298850195325332161233\right )}}\right ) + x \log \left (\frac {x^{8} + x^{3} + 3 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{x^{8} + x^{3} - 1}\right ) + 6 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{2 \, x} \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 (5 \, x^{8} + 3\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{{\left (x^{8} + x^{3} - 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 57.87, size = 611, normalized size = 5.46
method | result | size |
trager | \(\frac {3 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}}}{x}-3 \ln \left (\frac {5377901481395524539 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{8}-10267890788505543624 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{8}+4455082089494573732 x^{8}-487912174285505454 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -487912174285505454 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-12548436789922890591 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+11735249832780381501 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} x +11735249832780381501 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+15592787313231008062 x^{3}-5377901481395524539 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+10267890788505543624 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4455082089494573732}{x^{8}+x^{3}-1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {5377901481395524539 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{8}+13853158442769226650 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{8}+8475256961373702111 x^{8}+487912174285505454 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +487912174285505454 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+12548436789922890591 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+11897887224208883319 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} x +11897887224208883319 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+19775599576538638259 x^{3}-5377901481395524539 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-13853158442769226650 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-8475256961373702111}{x^{8}+x^{3}-1}\right )-\ln \left (\frac {5377901481395524539 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{8}-10267890788505543624 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{8}+4455082089494573732 x^{8}-487912174285505454 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -487912174285505454 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-12548436789922890591 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+11735249832780381501 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} x +11735249832780381501 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+15592787313231008062 x^{3}-5377901481395524539 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+10267890788505543624 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4455082089494573732}{x^{8}+x^{3}-1}\right )\) | \(611\) |
risch | \(\frac {3 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}}}{x}+\frac {\left (\ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{16}+x^{16}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{11}-5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{11}+4 x^{11}-3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} x^{9}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-2 x^{8}-6 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+4 x^{6}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {2}{3}} x^{2}-6 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} x^{4}+5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {2}{3}} x^{2}-4 x^{3}+3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} x -\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{\left (x^{8}+2 x^{3}-1\right ) \left (x^{8}+x^{3}-1\right )}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{16}-x^{16}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{11}+4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{11}+3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{9}-5 x^{11}+3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} x^{9}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+2 x^{8}+4 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+6 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-6 x^{6}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {2}{3}} x^{2}+6 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} x^{4}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +5 x^{3}-3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} x +\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{\left (x^{8}+2 x^{3}-1\right ) \left (x^{8}+x^{3}-1\right )}\right )\right ) \left (\left (x^{8}+2 x^{3}-1\right )^{2}\right )^{\frac {1}{3}}}{\left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}}}\) | \(787\) |
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 (5 \, x^{8} + 3\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{{\left (x^{8} + x^{3} - 1\right )} x^{2}}\,{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 (5\,x^8+3\right )\,{\left (x^8+2\,x^3-1\right )}^{1/3}}{x^2\,\left (x^8+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 (5 x^{8} + 3\right ) \sqrt [3]{x^{8} + 2 x^{3} - 1}}{x^{2} \left (x^{8} + x^{3} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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