Optimal. Leaf size=92 \[ \frac {1}{2} \log \left (\sqrt [3]{x^7+x^5+x}-x\right )+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^7+x^5+x}+x}\right )-\frac {1}{4} \log \left (x^2+\sqrt [3]{x^7+x^5+x} x+\left (x^7+x^5+x\right )^{2/3}\right ) \]
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Rubi [F] time = 2.39, 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^4+2 x^6\right ) \sqrt [3]{x+x^5+x^7}}{\left (1+x^4+x^6\right ) \left (1-x^2+x^4+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (-1+x^4+2 x^6\right ) \sqrt [3]{x+x^5+x^7}}{\left (1+x^4+x^6\right ) \left (1-x^2+x^4+x^6\right )} \, dx &=\frac {\sqrt [3]{x+x^5+x^7} \int \frac {\sqrt [3]{x} \left (-1+x^4+2 x^6\right )}{\left (1+x^4+x^6\right )^{2/3} \left (1-x^2+x^4+x^6\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (-1+x^{12}+2 x^{18}\right )}{\left (1+x^{12}+x^{18}\right )^{2/3} \left (1-x^6+x^{12}+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \left (-1+x^6+2 x^9\right )}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \left (\frac {2 x}{\left (1+x^6+x^9\right )^{2/3}}+\frac {x \left (-3+2 x^3-x^6\right )}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x \left (-3+2 x^3-x^6\right )}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}+\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^6+x^9\right )^{2/3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 x}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )}+\frac {2 x^4}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )}-\frac {x^7}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}+\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^6+x^9\right )^{2/3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ &=-\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}+\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^6+x^9\right )^{2/3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}+\frac {\left (3 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}-\frac {\left (9 \sqrt [3]{x+x^5+x^7}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^6+x^9\right )^{2/3} \left (1-x^3+x^6+x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+x^4+x^6}}\\ \end {align*}
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Mathematica [F] time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4+2 x^6\right ) \sqrt [3]{x+x^5+x^7}}{\left (1+x^4+x^6\right ) \left (1-x^2+x^4+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.19, size = 92, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (\sqrt [3]{x^7+x^5+x}-x\right )+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^7+x^5+x}+x}\right )-\frac {1}{4} \log \left (x^2+\sqrt [3]{x^7+x^5+x} x+\left (x^7+x^5+x\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 3.07, size = 127, normalized size = 1.38 \begin {gather*} \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{7} + x^{5} + x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{6} + x^{4} + x^{2} + 1\right )} + 2 \, \sqrt {3} {\left (x^{7} + x^{5} + x\right )}^{\frac {2}{3}}}{x^{6} + x^{4} - x^{2} + 1}\right ) + \frac {1}{4} \, \log \left (\frac {x^{6} + x^{4} - x^{2} + 3 \, {\left (x^{7} + x^{5} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{7} + x^{5} + x\right )}^{\frac {2}{3}} + 1}{x^{6} + x^{4} - x^{2} + 1}\right ) \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 (x^{7} + x^{5} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} + x^{4} - 1\right )}}{{\left (x^{6} + x^{4} - x^{2} + 1\right )} {\left (x^{6} + x^{4} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 8.38, size = 489, normalized size = 5.32 \begin {gather*} \frac {\ln \left (\frac {20906848965368 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}+44817676777674 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}+20906848965368 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}+6007957693876 x^{6}+44817676777674 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-62720546896104 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+6007957693876 x^{4}-35926743200058 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{7}+x^{5}+x \right )^{\frac {2}{3}}-31239270484854 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{7}+x^{5}+x \right )^{\frac {1}{3}} x +1441487941870 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+20906848965368 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+15619635242427 \left (x^{7}+x^{5}+x \right )^{\frac {2}{3}}-33583006842456 x \left (x^{7}+x^{5}+x \right )^{\frac {1}{3}}+1501989423469 x^{2}+44817676777674 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+6007957693876}{x^{6}+x^{4}-x^{2}+1}\right )}{2}+\RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {14898891271492 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}-25352315754176 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}+14898891271492 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{4}+4505968270407 x^{6}-25352315754176 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-44696673814476 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+4505968270407 x^{4}-35926743200058 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{7}+x^{5}+x \right )^{\frac {2}{3}}+67166013684912 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{7}+x^{5}+x \right )^{\frac {1}{3}} x -20785846002170 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+14898891271492 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-33583006842456 \left (x^{7}+x^{5}+x \right )^{\frac {2}{3}}+15619635242427 x \left (x^{7}+x^{5}+x \right )^{\frac {1}{3}}+6007957693876 x^{2}-25352315754176 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+4505968270407}{x^{6}+x^{4}-x^{2}+1}\right ) \end {gather*}
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 (x^{7} + x^{5} + x\right )}^{\frac {1}{3}} {\left (2 \, x^{6} + x^{4} - 1\right )}}{{\left (x^{6} + x^{4} - x^{2} + 1\right )} {\left (x^{6} + x^{4} + 1\right )}}\,{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 (2\,x^6+x^4-1\right )\,{\left (x^7+x^5+x\right )}^{1/3}}{\left (x^6+x^4+1\right )\,\left (x^6+x^4-x^2+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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