Optimal. Leaf size=150 \[ -\frac {\log \left (2^{2/3} \sqrt [3]{-x^8+2 x^3-2 x}-2 x\right )}{2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{-x^8+2 x^3-2 x}+x}\right )}{2^{2/3}}+\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{-x^8+2 x^3-2 x} x+\sqrt [3]{2} \left (-x^8+2 x^3-2 x\right )^{2/3}\right )}{2\ 2^{2/3}} \]
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Rubi [F] time = 14.57, 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 (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx &=\frac {\sqrt [3]{-2 x+2 x^3-x^8} \int \frac {\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7} \left (-4+5 x^7\right )}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\sqrt [3]{-2 x+2 x^3-x^8} \int \frac {\sqrt [3]{x} \left (-4+5 x^7\right )}{\left (-2+2 x^2-x^7\right )^{2/3} \left (2+x^7\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\sqrt [3]{-2 x+2 x^3-x^8} \int \left (\frac {5 \sqrt [3]{x}}{\left (-2+2 x^2-x^7\right )^{2/3}}-\frac {14 \sqrt [3]{x}}{\left (-2+2 x^2-x^7\right )^{2/3} \left (2+x^7\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (5 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (14 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-2+2 x^2-x^7\right )^{2/3} \left (2+x^7\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=\frac {\left (14 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \left (-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}-x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}+\sqrt [7]{-1} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}-(-1)^{2/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}+(-1)^{3/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}-(-1)^{4/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}+(-1)^{5/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}-(-1)^{6/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}-x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}+\sqrt [7]{-1} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}-(-1)^{2/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}+(-1)^{3/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}-(-1)^{4/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}+(-1)^{5/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}-(-1)^{6/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}-x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}+\sqrt [7]{-1} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}-(-1)^{2/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}+(-1)^{3/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}-(-1)^{4/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}+(-1)^{5/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}-(-1)^{6/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {\sqrt [7]{2}}{\left (-\sqrt [7]{2}-x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (-\frac {(-1)^{6/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {(-1)^{6/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}+\sqrt [7]{-1} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (\frac {(-1)^{5/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{5/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}-(-1)^{2/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (-\frac {(-1)^{4/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {(-1)^{4/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}+(-1)^{3/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (\frac {(-1)^{3/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{3/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}-(-1)^{4/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (-\frac {(-1)^{2/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {(-1)^{2/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}+(-1)^{5/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [7]{-1}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {\sqrt [7]{-2}}{\left (-\sqrt [7]{2}-(-1)^{6/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{-2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-2)^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}+(-1)^{5/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{2/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{3/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{4/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{5/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{6/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3\ 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}-x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{-1} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}-(-1)^{6/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{3/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}-(-1)^{4/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{4/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}+(-1)^{3/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{5/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}-(-1)^{2/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{6/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [7]{2}+\sqrt [7]{-1} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{-2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-2)^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (-\frac {1}{2}\right )^{2/21}}{3 \left (-(-1)^{2/21} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {\left (-\frac {1}{2}\right )^{2/21}}{3 \left (-(-1)^{2/21} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {\left (-\frac {1}{2}\right )^{2/21}}{3 \left (-(-1)^{2/21} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{2/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{3/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{4/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{5/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{6/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3\ 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{3\ 2^{2/21} \left (-\sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {1}{3\ 2^{2/21} \left (-\sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {1}{3\ 2^{2/21} \left (-\sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{-1} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt [21]{-1}}{3\ 2^{2/21} \left (\sqrt [21]{-2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {\sqrt [21]{-1}}{3\ 2^{2/21} \left (\sqrt [21]{-2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {\sqrt [21]{-1}}{3\ 2^{2/21} \left (\sqrt [21]{-2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{3/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt [7]{-1}}{3\ 2^{2/21} \left (\sqrt [7]{-1} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {\sqrt [7]{-1}}{3\ 2^{2/21} \left (\sqrt [7]{-1} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {\sqrt [7]{-1}}{3\ 2^{2/21} \left (\sqrt [7]{-1} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{4/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (\frac {(-1)^{4/21}}{3\ 2^{2/21} \left (-(-1)^{4/21} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{4/21}}{3\ 2^{2/21} \left (-(-1)^{4/21} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{4/21}}{3\ 2^{2/21} \left (-(-1)^{4/21} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{5/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (-\frac {(-1)^{5/21}}{3\ 2^{2/21} \left ((-1)^{5/21} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {(-1)^{5/21}}{3\ 2^{2/21} \left ((-1)^{5/21} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {(-1)^{5/21}}{3\ 2^{2/21} \left ((-1)^{5/21} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{6/7} 2^{2/7} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \left (\frac {(-1)^{2/7}}{3\ 2^{2/21} \left (-(-1)^{2/7} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{2/7}}{3\ 2^{2/21} \left (-(-1)^{2/7} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{2/7}}{3\ 2^{2/21} \left (-(-1)^{2/7} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ &=-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{-2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-2)^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [21]{-2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-2)^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [21]{-2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-2)^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [21]{-2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{2/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{3/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{4/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 (-1)^{5/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (3 (-1)^{6/7} \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{-1} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-(-1)^{2/7} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{-1} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-(-1)^{2/7} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{-1} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-(-1)^{2/7} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{8/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-(-1)^{2/21} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{8/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-(-1)^{2/21} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{8/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-(-1)^{2/21} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{4/7} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [7]{-1} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{4/7} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [7]{-1} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{4/7} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt [7]{-1} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{16/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-(-1)^{4/21} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{16/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-(-1)^{4/21} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{16/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-(-1)^{4/21} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{20/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left ((-1)^{5/21} \sqrt [21]{2}-x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{20/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left ((-1)^{5/21} \sqrt [21]{2}+\sqrt [3]{-1} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left ((-1)^{20/21} 2^{4/21} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \operatorname {Subst}\left (\int \frac {1}{\left ((-1)^{5/21} \sqrt [21]{2}-(-1)^{2/3} x\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}\\ \end {align*}
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Mathematica [F] time = 3.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.41, size = 150, normalized size = 1.00 \begin {gather*} -\frac {\log \left (2^{2/3} \sqrt [3]{-x^8+2 x^3-2 x}-2 x\right )}{2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{-x^8+2 x^3-2 x}+x}\right )}{2^{2/3}}+\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{-x^8+2 x^3-2 x} x+\sqrt [3]{2} \left (-x^8+2 x^3-2 x\right )^{2/3}\right )}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 7.94, size = 390, normalized size = 2.60 \begin {gather*} \frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (-\frac {4^{\frac {1}{6}} \sqrt {3} {\left (6 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{15} - 18 \, x^{10} + 4 \, x^{8} + 36 \, x^{5} - 36 \, x^{3} + 4 \, x\right )} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} - 12 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{14} - 6 \, x^{9} + 4 \, x^{7} - 12 \, x^{2} + 4\right )} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} {\left (x^{21} - 36 \, x^{16} + 6 \, x^{14} + 180 \, x^{11} - 144 \, x^{9} + 12 \, x^{7} - 216 \, x^{6} + 360 \, x^{4} - 144 \, x^{2} + 8\right )}\right )}}{6 \, {\left (x^{21} + 6 \, x^{14} - 108 \, x^{11} + 12 \, x^{7} + 216 \, x^{6} - 216 \, x^{4} + 8\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {2}{3}} {\left (x^{7} - 6 \, x^{2} + 2\right )} + 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{14} - 18 \, x^{9} + 4 \, x^{7} + 36 \, x^{4} - 36 \, x^{2} + 4\right )} + 24 \, {\left (x^{8} - 3 \, x^{3} + 2 \, x\right )} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}}}{x^{14} + 4 \, x^{7} + 4}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} x + 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{7} + 2\right )} + 6 \, {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {2}{3}}}{x^{7} + 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} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} - 4\right )}}{{\left (x^{7} - 2 \, x^{2} + 2\right )} {\left (x^{7} + 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (5 x^{7}-4\right ) \left (-x^{8}+2 x^{3}-2 x \right )^{\frac {1}{3}}}{\left (x^{7}+2\right ) \left (x^{7}-2 x^{2}+2\right )}\, dx \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^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} - 4\right )}}{{\left (x^{7} - 2 \, x^{2} + 2\right )} {\left (x^{7} + 2\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 (5\,x^7-4\right )\,{\left (-x^8+2\,x^3-2\,x\right )}^{1/3}}{\left (x^7+2\right )\,\left (x^7-2\,x^2+2\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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