Optimal. Leaf size=225 \[ -\log \left (\sqrt [3]{x^4+2 x^3+x}-x\right )+\sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{x^4+2 x^3+x}-2 x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+2 x^3+x}+x}\right )+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+2 x^3+x}+x}\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^4+2 x^3+x} x+\left (x^4+2 x^3+x\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^4+2 x^3+x} x+\sqrt [3]{2} \left (x^4+2 x^3+x\right )^{2/3}\right )}{2^{2/3}} \]
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Rubi [F] time = 6.27, 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+x^3\right ) \sqrt [3]{x+2 x^3+x^4}}{\left (1+x^3\right ) \left (1+x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+2 x^3+x^4}}{\left (1+x^3\right ) \left (1+x^2+x^3\right )} \, dx &=\frac {\sqrt [3]{x+2 x^3+x^4} \int \frac {\sqrt [3]{x} \left (-2+x^3\right ) \sqrt [3]{1+2 x^2+x^3}}{\left (1+x^3\right ) \left (1+x^2+x^3\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\sqrt [3]{x+2 x^3+x^4} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{-1-x}+\frac {\sqrt [3]{x} (1+x) \sqrt [3]{1+2 x^2+x^3}}{1-x+x^2}+\frac {(-2-3 x) \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{1+x^2+x^3}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\sqrt [3]{x+2 x^3+x^4} \int \frac {\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{-1-x} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\sqrt [3]{x+2 x^3+x^4} \int \frac {\sqrt [3]{x} (1+x) \sqrt [3]{1+2 x^2+x^3}}{1-x+x^2} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\sqrt [3]{x+2 x^3+x^4} \int \frac {(-2-3 x) \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{1+x^2+x^3} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\sqrt [3]{x+2 x^3+x^4} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{-1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{-1+i \sqrt {3}+2 x}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{-1-x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (-2-3 x^3\right ) \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (-\sqrt [3]{1+2 x^6+x^9}-\frac {\sqrt [3]{1+2 x^6+x^9}}{-1-x^3}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {2 x^3 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9}-\frac {3 x^6 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{-1-i \sqrt {3}+2 x} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}{-1+i \sqrt {3}+2 x} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{-1-x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (9 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{-1-i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{-1+i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt [3]{1+2 x^6+x^9}}{3 (1+x)}+\frac {(-2+x) \sqrt [3]{1+2 x^6+x^9}}{3 \left (1-x+x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (9 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2} \sqrt [3]{1+2 x^6+x^9}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{1+2 x^6+x^9}}{2 \left (-1-i \sqrt {3}+2 x^3\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2} \sqrt [3]{1+2 x^6+x^9}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{1+2 x^6+x^9}}{2 \left (-1+i \sqrt {3}+2 x^3\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\sqrt [3]{x+2 x^3+x^4} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\sqrt [3]{x+2 x^3+x^4} \operatorname {Subst}\left (\int \frac {(-2+x) \sqrt [3]{1+2 x^6+x^9}}{1-x+x^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (9 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{-1-i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{-1+i \sqrt {3}+2 x^3} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\sqrt [3]{x+2 x^3+x^4} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\sqrt [3]{x+2 x^3+x^4} \operatorname {Subst}\left (\int \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{1+2 x^6+x^9}}{-1-i \sqrt {3}+2 x}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{1+2 x^6+x^9}}{-1+i \sqrt {3}+2 x}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (9 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+2 x^6+x^9}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+2 x^6+x^9}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1-i \sqrt {3}} \sqrt [3]{1+2 x^6+x^9}}{3 \left (-1+i \sqrt {3}\right ) \left (\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{1+2 x^6+x^9}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x\right )}+\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{1+2 x^6+x^9}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x\right )}+\frac {\sqrt [3]{1+i \sqrt {3}} \sqrt [3]{1+2 x^6+x^9}}{3 \left (-1-i \sqrt {3}\right ) \left (\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ &=\frac {\sqrt [3]{x+2 x^3+x^4} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{1+x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (3 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (9 \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [3]{1+2 x^6+x^9}}{1+x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{-1+i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{\sqrt [3]{1+i \sqrt {3}}+\sqrt [3]{-2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{\sqrt [3]{1+i \sqrt {3}}-\sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [3]{1+i \sqrt {3}} \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{\sqrt [3]{1+i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{-1-i \sqrt {3}+2 x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}+\frac {\left (3 \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \sqrt [3]{1+2 x^6+x^9} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\sqrt [3]{1-i \sqrt {3}} \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{\sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{-2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\sqrt [3]{1-i \sqrt {3}} \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{\sqrt [3]{1-i \sqrt {3}}-\sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}-\frac {\left (\sqrt [3]{1-i \sqrt {3}} \left (1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+2 x^6+x^9}}{\sqrt [3]{1-i \sqrt {3}}-(-1)^{2/3} \sqrt [3]{2} x} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2+x^3}}\\ \end {align*}
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Mathematica [F] time = 2.52, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+2 x^3+x^4}}{\left (1+x^3\right ) \left (1+x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.61, size = 225, normalized size = 1.00 \begin {gather*} -\log \left (\sqrt [3]{x^4+2 x^3+x}-x\right )+\sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{x^4+2 x^3+x}-2 x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^4+2 x^3+x}+x}\right )+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+2 x^3+x}+x}\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^4+2 x^3+x} x+\left (x^4+2 x^3+x\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^4+2 x^3+x} x+\sqrt [3]{2} \left (x^4+2 x^3+x\right )^{2/3}\right )}{2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{{\left (x^{3} + x^{2} + 1\right )} {\left (x^{3} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 31.24, size = 2504, normalized size = 11.13 \begin {gather*} \text {Expression too large to display} \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^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{{\left (x^{3} + x^{2} + 1\right )} {\left (x^{3} + 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 = -0.00 \begin {gather*} \int \frac {\left (x^3-2\right )\,{\left (x^4+2\,x^3+x\right )}^{1/3}}{\left (x^3+1\right )\,\left (x^3+x^2+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 {\sqrt [3]{x \left (x^{3} + 2 x^{2} + 1\right )} \left (x^{3} - 2\right )}{\left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{3} + x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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