Optimal. Leaf size=95 \[ -\frac {3 \sqrt [3]{x^3+x+1}}{x}-\log \left (\sqrt [3]{x^3+x+1}-x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x+1}+x}\right )+\frac {1}{2} \log \left (\sqrt [3]{x^3+x+1} x+\left (x^3+x+1\right )^{2/3}+x^2\right ) \]
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Rubi [F] time = 1.47, 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 {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx &=\int \left (\frac {3 \sqrt [3]{1+x+x^3}}{x^2}-\frac {\sqrt [3]{1+x+x^3}}{x}+\frac {\sqrt [3]{1+x+x^3}}{1+x}\right ) \, dx\\ &=3 \int \frac {\sqrt [3]{1+x+x^3}}{x^2} \, dx-\int \frac {\sqrt [3]{1+x+x^3}}{x} \, dx+\int \frac {\sqrt [3]{1+x+x^3}}{1+x} \, dx\\ &=-\frac {\sqrt [3]{1+x+x^3} \int \frac {\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}{x} \, dx}{\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}+\frac {\sqrt [3]{1+x+x^3} \int \frac {\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}{1+x} \, dx}{\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}+\frac {\left (3 \sqrt [3]{1+x+x^3}\right ) \int \frac {\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}{x^2} \, dx}{\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}\\ \end {align*}
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Mathematica [F] time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.26, size = 95, normalized size = 1.00 \begin {gather*} -\frac {3 \sqrt [3]{1+x+x^3}}{x}-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x+x^3}}\right )-\log \left (-x+\sqrt [3]{1+x+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x+x^3}+\left (1+x+x^3\right )^{2/3}\right ) \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^{3} + x + 1\right )}^{\frac {1}{3}} {\left (2 \, x + 3\right )}}{{\left (x + 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 6.98, size = 389, normalized size = 4.09
method | result | size |
trager | \(-\frac {3 \left (x^{3}+x +1\right )^{\frac {1}{3}}}{x}-\ln \left (\frac {18990 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x^{3}-37665 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}+x +1\right )^{\frac {2}{3}} x +3357 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+21648 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x^{3}-9495 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x +2238 \left (x^{3}+x +1\right )^{\frac {2}{3}} x +22872 \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-16670 x^{3}-9495 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2}+30321 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x +30321 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )-13336 x -13336}{1+x}\right )+\frac {3 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \ln \left (-\frac {7974 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x^{3}-75330 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}+x +1\right )^{\frac {2}{3}} x +68616 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+1398 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x^{3}-3987 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x +45744 \left (x^{3}+x +1\right )^{\frac {2}{3}} x +4476 \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-46676 x^{3}-3987 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2}-17976 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x -17976 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )-20004 x -20004}{1+x}\right )}{2}\) | \(389\) |
risch | \(-\frac {3 \left (x^{3}+x +1\right )^{\frac {1}{3}}}{x}+\frac {\left (-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{6}+4 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}+4 x^{6}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}-6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {2}{3}} x^{2}+6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{4}+6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+12 x^{2} \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {2}{3}}+8 x^{4}+2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{2}+8 x^{3}+4 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x +4 x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )+8 x +4}{\left (1+x \right ) \left (x^{3}+x +1\right )}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{6}-2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}-6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{4}-8 x^{6}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}+6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {2}{3}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}+12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{4}-6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}-12 x^{4}-6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x -2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{2}-12 x^{3}-4 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x -4 x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-8 x -4}{\left (1+x \right ) \left (x^{3}+x +1\right )}\right )}{2}\right ) \left (\left (x^{3}+x +1\right )^{2}\right )^{\frac {1}{3}}}{\left (x^{3}+x +1\right )^{\frac {2}{3}}}\) | \(753\) |
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^{3} + x + 1\right )}^{\frac {1}{3}} {\left (2 \, x + 3\right )}}{{\left (x + 1\right )} x^{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 = -0.01 \begin {gather*} \int \frac {\left (2\,x+3\right )\,{\left (x^3+x+1\right )}^{1/3}}{x^2\,\left (x+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 {\left (2 x + 3\right ) \sqrt [3]{x^{3} + x + 1}}{x^{2} \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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