Optimal. Leaf size=238 \[ \frac {\log \left (2 \sqrt [3]{x^8+x^5-x^3+x}+2^{2/3} \sqrt [3]{3} x\right )}{6 \sqrt [3]{2} 3^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^8+x^5-x^3+x}}{2^{2/3} \sqrt [3]{3} x-\sqrt [3]{x^8+x^5-x^3+x}}\right )}{6 \sqrt [3]{2} \sqrt [6]{3}}-\frac {\log \left (\sqrt [3]{2} 3^{2/3} x^2-2^{2/3} \sqrt [3]{3} \sqrt [3]{x^8+x^5-x^3+x} x+2 \left (x^8+x^5-x^3+x\right )^{2/3}\right )}{12 \sqrt [3]{2} 3^{2/3}}-\frac {\sqrt [3]{x^8+x^5-x^3+x} x}{2 \left (2 x^7+2 x^4+x^2+2\right )} \]
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Rubi [F] time = 2.92, 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 (-2+2 x^4+5 x^7\right ) \sqrt [3]{x-x^3+x^5+x^8}}{\left (2+x^2+2 x^4+2 x^7\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (-2+2 x^4+5 x^7\right ) \sqrt [3]{x-x^3+x^5+x^8}}{\left (2+x^2+2 x^4+2 x^7\right )^2} \, dx &=\frac {\sqrt [3]{x-x^3+x^5+x^8} \int \frac {\sqrt [3]{x} \sqrt [3]{1-x^2+x^4+x^7} \left (-2+2 x^4+5 x^7\right )}{\left (2+x^2+2 x^4+2 x^7\right )^2} \, dx}{\sqrt [3]{x} \sqrt [3]{1-x^2+x^4+x^7}}\\ &=\frac {\left (3 \sqrt [3]{x-x^3+x^5+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^6+x^{12}+x^{21}} \left (-2+2 x^{12}+5 x^{21}\right )}{\left (2+x^6+2 x^{12}+2 x^{21}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^2+x^4+x^7}}\\ &=\frac {\left (3 \sqrt [3]{x-x^3+x^5+x^8}\right ) \operatorname {Subst}\left (\int \left (-\frac {x^3 \left (14+5 x^6+6 x^{12}\right ) \sqrt [3]{1-x^6+x^{12}+x^{21}}}{2 \left (2+x^6+2 x^{12}+2 x^{21}\right )^2}+\frac {5 x^3 \sqrt [3]{1-x^6+x^{12}+x^{21}}}{2 \left (2+x^6+2 x^{12}+2 x^{21}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^2+x^4+x^7}}\\ &=-\frac {\left (3 \sqrt [3]{x-x^3+x^5+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (14+5 x^6+6 x^{12}\right ) \sqrt [3]{1-x^6+x^{12}+x^{21}}}{\left (2+x^6+2 x^{12}+2 x^{21}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^2+x^4+x^7}}+\frac {\left (15 \sqrt [3]{x-x^3+x^5+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^6+x^{12}+x^{21}}}{2+x^6+2 x^{12}+2 x^{21}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^2+x^4+x^7}}\\ &=-\frac {\left (3 \sqrt [3]{x-x^3+x^5+x^8}\right ) \operatorname {Subst}\left (\int \left (\frac {14 x^3 \sqrt [3]{1-x^6+x^{12}+x^{21}}}{\left (2+x^6+2 x^{12}+2 x^{21}\right )^2}+\frac {5 x^9 \sqrt [3]{1-x^6+x^{12}+x^{21}}}{\left (2+x^6+2 x^{12}+2 x^{21}\right )^2}+\frac {6 x^{15} \sqrt [3]{1-x^6+x^{12}+x^{21}}}{\left (2+x^6+2 x^{12}+2 x^{21}\right )^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^2+x^4+x^7}}+\frac {\left (15 \sqrt [3]{x-x^3+x^5+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^6+x^{12}+x^{21}}}{2+x^6+2 x^{12}+2 x^{21}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^2+x^4+x^7}}\\ &=-\frac {\left (15 \sqrt [3]{x-x^3+x^5+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^9 \sqrt [3]{1-x^6+x^{12}+x^{21}}}{\left (2+x^6+2 x^{12}+2 x^{21}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^2+x^4+x^7}}+\frac {\left (15 \sqrt [3]{x-x^3+x^5+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^6+x^{12}+x^{21}}}{2+x^6+2 x^{12}+2 x^{21}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1-x^2+x^4+x^7}}-\frac {\left (9 \sqrt [3]{x-x^3+x^5+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^{15} \sqrt [3]{1-x^6+x^{12}+x^{21}}}{\left (2+x^6+2 x^{12}+2 x^{21}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^2+x^4+x^7}}-\frac {\left (21 \sqrt [3]{x-x^3+x^5+x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1-x^6+x^{12}+x^{21}}}{\left (2+x^6+2 x^{12}+2 x^{21}\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^2+x^4+x^7}}\\ \end {align*}
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Mathematica [F] time = 1.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2+2 x^4+5 x^7\right ) \sqrt [3]{x-x^3+x^5+x^8}}{\left (2+x^2+2 x^4+2 x^7\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 5.37, size = 238, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt [3]{x-x^3+x^5+x^8}}{2 \left (2+x^2+2 x^4+2 x^7\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-x^3+x^5+x^8}}{2^{2/3} \sqrt [3]{3} x-\sqrt [3]{x-x^3+x^5+x^8}}\right )}{6 \sqrt [3]{2} \sqrt [6]{3}}+\frac {\log \left (2^{2/3} \sqrt [3]{3} x+2 \sqrt [3]{x-x^3+x^5+x^8}\right )}{6 \sqrt [3]{2} 3^{2/3}}-\frac {\log \left (\sqrt [3]{2} 3^{2/3} x^2-2^{2/3} \sqrt [3]{3} x \sqrt [3]{x-x^3+x^5+x^8}+2 \left (x-x^3+x^5+x^8\right )^{2/3}\right )}{12 \sqrt [3]{2} 3^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 60.13, size = 635, normalized size = 2.67 \begin {gather*} -\frac {6 \cdot 18^{\frac {1}{6}} \sqrt {6} {\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\right )} \arctan \left (-\frac {18^{\frac {1}{6}} {\left (6 \cdot 18^{\frac {2}{3}} \sqrt {6} {\left (4 \, x^{15} + 8 \, x^{12} - 50 \, x^{10} + 4 \, x^{9} + 8 \, x^{8} - 50 \, x^{7} + 63 \, x^{5} - 50 \, x^{3} + 4 \, x\right )} {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {1}{3}} - 72 \, \sqrt {6} {\left (2 \, x^{14} + 4 \, x^{11} - 7 \, x^{9} + 2 \, x^{8} + 4 \, x^{7} - 7 \, x^{6} - 7 \, x^{2} + 2\right )} {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {2}{3}} + 18^{\frac {1}{3}} \sqrt {6} {\left (8 \, x^{21} + 24 \, x^{18} - 204 \, x^{16} + 24 \, x^{15} + 24 \, x^{14} - 408 \, x^{13} + 8 \, x^{12} + 648 \, x^{11} - 204 \, x^{10} - 408 \, x^{9} + 624 \, x^{8} + 24 \, x^{7} - 785 \, x^{6} + 624 \, x^{4} - 204 \, x^{2} + 8\right )}\right )}}{18 \, {\left (8 \, x^{21} + 24 \, x^{18} + 12 \, x^{16} + 24 \, x^{15} + 24 \, x^{14} + 24 \, x^{13} + 8 \, x^{12} - 432 \, x^{11} + 12 \, x^{10} + 24 \, x^{9} - 456 \, x^{8} + 24 \, x^{7} + 511 \, x^{6} - 456 \, x^{4} + 12 \, x^{2} + 8\right )}}\right ) + 18^{\frac {2}{3}} {\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\right )} \log \left (-\frac {36 \cdot 18^{\frac {1}{3}} {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {2}{3}} {\left (x^{7} + x^{4} - 4 \, x^{2} + 1\right )} - 18^{\frac {2}{3}} {\left (4 \, x^{14} + 8 \, x^{11} - 50 \, x^{9} + 4 \, x^{8} + 8 \, x^{7} - 50 \, x^{6} + 63 \, x^{4} - 50 \, x^{2} + 4\right )} - 54 \, {\left (4 \, x^{8} + 4 \, x^{5} - 7 \, x^{3} + 4 \, x\right )} {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {1}{3}}}{4 \, x^{14} + 8 \, x^{11} + 4 \, x^{9} + 4 \, x^{8} + 8 \, x^{7} + 4 \, x^{6} + 9 \, x^{4} + 4 \, x^{2} + 4}\right ) - 2 \cdot 18^{\frac {2}{3}} {\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\right )} \log \left (\frac {3 \cdot 18^{\frac {2}{3}} {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {1}{3}} x + 18^{\frac {1}{3}} {\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\right )} + 18 \, {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {2}{3}}}{2 \, x^{7} + 2 \, x^{4} + x^{2} + 2}\right ) + 324 \, {\left (x^{8} + x^{5} - x^{3} + x\right )}^{\frac {1}{3}} x}{648 \, {\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\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^{8} + x^{5} - x^{3} + x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} + 2 \, x^{4} - 2\right )}}{{\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (5 x^{7}+2 x^{4}-2\right ) \left (x^{8}+x^{5}-x^{3}+x \right )^{\frac {1}{3}}}{\left (2 x^{7}+2 x^{4}+x^{2}+2\right )^{2}}\, dx\]
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^{8} + x^{5} - x^{3} + x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} + 2 \, x^{4} - 2\right )}}{{\left (2 \, x^{7} + 2 \, x^{4} + x^{2} + 2\right )}^{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 = -0.00 \begin {gather*} \int \frac {\left (5\,x^7+2\,x^4-2\right )\,{\left (x^8+x^5-x^3+x\right )}^{1/3}}{{\left (2\,x^7+2\,x^4+x^2+2\right )}^2} \,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 {\sqrt [3]{x \left (x + 1\right ) \left (x^{6} - x^{5} + x^{4} - x + 1\right )} \left (5 x^{7} + 2 x^{4} - 2\right )}{\left (2 x^{7} + 2 x^{4} + x^{2} + 2\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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