Optimal. Leaf size=105 \[ -\sqrt {3} \text {ArcTan}\left (\frac {\frac {2}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {\sqrt [3]{1+x+x^2}}{\sqrt {3}}}{\sqrt [3]{1+x+x^2}}\right )+\log \left (-1+x+\sqrt [3]{1+x+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2+(1-x) \sqrt [3]{1+x+x^2}+\left (1+x+x^2\right )^{2/3}\right ) \]
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Rubi [F]
time = 0.32, 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 {(2+x)^2}{x \left (4-2 x+x^2\right ) \sqrt [3]{1+x+x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(2+x)^2}{x \left (4-2 x+x^2\right ) \sqrt [3]{1+x+x^2}} \, dx &=\int \left (\frac {1}{x \sqrt [3]{1+x+x^2}}+\frac {6}{\left (4-2 x+x^2\right ) \sqrt [3]{1+x+x^2}}\right ) \, dx\\ &=6 \int \frac {1}{\left (4-2 x+x^2\right ) \sqrt [3]{1+x+x^2}} \, dx+\int \frac {1}{x \sqrt [3]{1+x+x^2}} \, dx\\ &=6 \int \frac {1}{\left (4-2 x+x^2\right ) \sqrt [3]{1+x+x^2}} \, dx-\frac {\left (\sqrt [3]{\frac {1-i \sqrt {3}+2 x}{x}} \sqrt [3]{\frac {1+i \sqrt {3}+2 x}{x}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{x} \sqrt [3]{1+\frac {1}{2} \left (1-i \sqrt {3}\right ) x} \sqrt [3]{1+\frac {1}{2} \left (1+i \sqrt {3}\right ) x}} \, dx,x,\frac {1}{x}\right )}{2^{2/3} \left (\frac {1}{x}\right )^{2/3} \sqrt [3]{1+x+x^2}}\\ &=-\frac {3 \sqrt [3]{\frac {1-i \sqrt {3}+2 x}{x}} \sqrt [3]{\frac {1+i \sqrt {3}+2 x}{x}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};-\frac {1-i \sqrt {3}}{2 x},-\frac {1+i \sqrt {3}}{2 x}\right )}{2\ 2^{2/3} \sqrt [3]{1+x+x^2}}+6 \int \frac {1}{\left (4-2 x+x^2\right ) \sqrt [3]{1+x+x^2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 92, normalized size = 0.88 \begin {gather*} -\sqrt {3} \text {ArcTan}\left (\frac {2-2 x+\sqrt [3]{1+x+x^2}}{\sqrt {3} \sqrt [3]{1+x+x^2}}\right )+\log \left (-1+x+\sqrt [3]{1+x+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2-(-1+x) \sqrt [3]{1+x+x^2}+\left (1+x+x^2\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.73, size = 575, normalized size = 5.48
method | result | size |
trager | \(\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+x +1\right )^{\frac {2}{3}} x -\left (x^{2}+x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+x +1\right )^{\frac {2}{3}}+2 x \left (x^{2}+x +1\right )^{\frac {2}{3}}+2 \left (x^{2}+x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2 \left (x^{2}+x +1\right )^{\frac {1}{3}} x^{2}-3 x^{2} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+x^{3}-2 \left (x^{2}+x +1\right )^{\frac {2}{3}}-\left (x^{2}+x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+4 x \left (x^{2}+x +1\right )^{\frac {1}{3}}+3 x \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-4 x^{2}-2 \left (x^{2}+x +1\right )^{\frac {1}{3}}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2 x -2}{\left (x^{2}-2 x +4\right ) x}\right )-\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{2}+x +1\right )^{\frac {2}{3}}-3 \left (x^{2}+x +1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +5 x^{2} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-3 \left (x^{2}+x +1\right )^{\frac {2}{3}}+6 x \left (x^{2}+x +1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-7 x \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 x^{2}-3 \left (x^{2}+x +1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 x -2}{\left (x^{2}-2 x +4\right ) x}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{2}+x +1\right )^{\frac {2}{3}}-3 \left (x^{2}+x +1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +5 x^{2} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-3 \left (x^{2}+x +1\right )^{\frac {2}{3}}+6 x \left (x^{2}+x +1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-7 x \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 x^{2}-3 \left (x^{2}+x +1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 x -2}{\left (x^{2}-2 x +4\right ) x}\right )\) | \(575\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.81, size = 139, normalized size = 1.32 \begin {gather*} -\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{2} + x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} + \sqrt {3} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}}{x^{3} - 11 \, x^{2} - 5 \, x - 9}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} - 2 \, x^{2} + 3 \, {\left (x^{2} + x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 3 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} + 4 \, x}{x^{3} - 2 \, x^{2} + 4 \, x}\right ) \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 (x + 2\right )^{2}}{x \left (x^{2} - 2 x + 4\right ) \sqrt [3]{x^{2} + x + 1}}\, dx \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*} \text {could not integrate} \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 (x+2\right )}^2}{x\,\left (x^2-2\,x+4\right )\,{\left (x^2+x+1\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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