Optimal. Leaf size=423 \[ \frac {\sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}} \left (-3 x^4+x^2-x-1\right )}{x^4}+\frac {5}{3} \log \left (\sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}-1\right )-\sqrt [3]{6} \log \left (6^{2/3} \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}-3\right )-\frac {5}{6} \log \left (\left (\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}\right )^{2/3}+\sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}+1\right )+\frac {\sqrt [3]{3} \log \left (2 \sqrt [3]{6} \left (\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}\right )^{2/3}+6^{2/3} \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}+3\right )}{2^{2/3}}+\sqrt [3]{2} 3^{5/6} \tan ^{-1}\left (\frac {2\ 2^{2/3} \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{3^{5/6}}+\frac {1}{\sqrt {3}}\right )-\frac {5 \tan ^{-1}\left (\frac {2 \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [F] time = 10.03, 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 (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx &=\frac {\left (\sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{1+x-x^2+2 x^4}}{x^5 \left (-1-x+x^2+x^4\right ) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}\\ &=\frac {\left (\sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \left (-\frac {2 \sqrt [3]{1+x-x^2+2 x^4}}{(-1+x) \sqrt [3]{1+x-x^2+3 x^4}}+\frac {4 \sqrt [3]{1+x-x^2+2 x^4}}{x^5 \sqrt [3]{1+x-x^2+3 x^4}}+\frac {3 \sqrt [3]{1+x-x^2+2 x^4}}{x^4 \sqrt [3]{1+x-x^2+3 x^4}}-\frac {2 \sqrt [3]{1+x-x^2+2 x^4}}{x^3 \sqrt [3]{1+x-x^2+3 x^4}}+\frac {8 \sqrt [3]{1+x-x^2+2 x^4}}{x \sqrt [3]{1+x-x^2+3 x^4}}-\frac {2 \left (2+2 x+3 x^2\right ) \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}}\right ) \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}\\ &=-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{(-1+x) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^3 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\left (2+2 x+3 x^2\right ) \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (3 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^4 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (4 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^5 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (8 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}\\ &=-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{(-1+x) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^3 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \left (\frac {2 \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}}+\frac {2 x \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}}+\frac {3 x^2 \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}}\right ) \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (3 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^4 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (4 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^5 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (8 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}\\ &=-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{(-1+x) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^3 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (3 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^4 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (4 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x^5 \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (4 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (4 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {x \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}-\frac {\left (6 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {x^2 \sqrt [3]{1+x-x^2+2 x^4}}{\left (1+2 x+x^2+x^3\right ) \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}+\frac {\left (8 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}} \sqrt [3]{1+x-x^2+3 x^4}\right ) \int \frac {\sqrt [3]{1+x-x^2+2 x^4}}{x \sqrt [3]{1+x-x^2+3 x^4}} \, dx}{\sqrt [3]{1+x-x^2+2 x^4}}\\ \end {align*}
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Mathematica [F] time = 1.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.72, size = 423, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}} \left (-3 x^4+x^2-x-1\right )}{x^4}+\frac {5}{3} \log \left (\sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}-1\right )-\sqrt [3]{6} \log \left (6^{2/3} \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}-3\right )-\frac {5}{6} \log \left (\left (\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}\right )^{2/3}+\sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}+1\right )+\frac {\sqrt [3]{3} \log \left (2 \sqrt [3]{6} \left (\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}\right )^{2/3}+6^{2/3} \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}+3\right )}{2^{2/3}}+\sqrt [3]{2} 3^{5/6} \tan ^{-1}\left (\frac {2\ 2^{2/3} \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{3^{5/6}}+\frac {1}{\sqrt {3}}\right )-\frac {5 \tan ^{-1}\left (\frac {2 \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 63.51, size = 960, normalized size = 2.27
result too large to display
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} - x^{2} + x + 1\right )} {\left (2 \, x^{2} - 3 \, x - 4\right )} \left (\frac {2 \, x^{4} - x^{2} + x + 1}{3 \, x^{4} - x^{2} + x + 1}\right )^{\frac {1}{3}}}{{\left (x^{4} + x^{2} - x - 1\right )} x^{5}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.82, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 x^{2}-3 x -4\right ) \left (x^{4}-x^{2}+x +1\right ) \left (\frac {2 x^{4}-x^{2}+x +1}{3 x^{4}-x^{2}+x +1}\right )^{\frac {1}{3}}}{x^{5} \left (x^{4}+x^{2}-x -1\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^{4} - x^{2} + x + 1\right )} {\left (2 \, x^{2} - 3 \, x - 4\right )} \left (\frac {2 \, x^{4} - x^{2} + x + 1}{3 \, x^{4} - x^{2} + x + 1}\right )^{\frac {1}{3}}}{{\left (x^{4} + x^{2} - x - 1\right )} x^{5}}\,{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 (\frac {2\,x^4-x^2+x+1}{3\,x^4-x^2+x+1}\right )}^{1/3}\,\left (-2\,x^2+3\,x+4\right )\,\left (x^4-x^2+x+1\right )}{x^5\,\left (-x^4-x^2+x+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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