Integrand size = 27, antiderivative size = 97 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+2 x^3\right )} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (-4+11 x^3\right )}{10 x^5}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )+\log \left (x+\sqrt [3]{1+x^3}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {594, 597, 12, 384} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+2 x^3\right )} \, dx=-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (2 x^3+1\right )+\frac {3}{2} \log \left (-\sqrt [3]{x^3+1}-x\right )-\frac {2 \left (x^3+1\right )^{2/3}}{5 x^5}+\frac {11 \left (x^3+1\right )^{2/3}}{10 x^2} \]
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Rule 12
Rule 384
Rule 594
Rule 597
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {1}{5} \int \frac {-11-7 x^3}{x^3 \sqrt [3]{1+x^3} \left (1+2 x^3\right )} \, dx \\ & = -\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {11 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {1}{10} \int -\frac {30}{\sqrt [3]{1+x^3} \left (1+2 x^3\right )} \, dx \\ & = -\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {11 \left (1+x^3\right )^{2/3}}{10 x^2}+3 \int \frac {1}{\sqrt [3]{1+x^3} \left (1+2 x^3\right )} \, dx \\ & = -\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}+\frac {11 \left (1+x^3\right )^{2/3}}{10 x^2}-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {1}{2} \log \left (1+2 x^3\right )+\frac {3}{2} \log \left (-x-\sqrt [3]{1+x^3}\right ) \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+2 x^3\right )} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (-4+11 x^3\right )}{10 x^5}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )+\log \left (x+\sqrt [3]{1+x^3}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
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Time = 6.42 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.03
method | result | size |
pseudoelliptic | \(\frac {10 \ln \left (\frac {x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (11 x^{3}-4\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+5 x^{5} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )-\ln \left (\frac {x^{2}-x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{10 x^{5}}\) | \(100\) |
risch | \(\frac {11 x^{6}+7 x^{3}-4}{10 x^{5} \left (x^{3}+1\right )^{\frac {1}{3}}}+\ln \left (-\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +6 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-x \left (x^{3}+1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-1}{2 x^{3}+1}\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -3 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2 x \left (x^{3}+1\right )^{\frac {2}{3}}+x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{2 x^{3}+1}\right )\) | \(258\) |
trager | \(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (11 x^{3}-4\right )}{10 x^{5}}+\ln \left (\frac {-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +63 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+21 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-15 x \left (x^{3}+1\right )^{\frac {2}{3}}+6 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-5 x^{3}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-10}{2 x^{3}+1}\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -45 \left (x^{3}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+24 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+21 x \left (x^{3}+1\right )^{\frac {2}{3}}+6 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-4 x^{3}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+21 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{2 x^{3}+1}\right )\) | \(327\) |
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Time = 0.50 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.28 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+2 x^3\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} + 1\right )}}{7 \, x^{3} - 1}\right ) - 5 \, x^{5} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{3} + 1}\right ) - {\left (11 \, x^{3} - 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{10 \, x^{5}} \]
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\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+2 x^3\right )} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} + 2\right )}{x^{6} \cdot \left (2 x^{3} + 1\right )}\, dx \]
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\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} + 1\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{3} + 1\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+2 x^3\right )} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^3+2\right )}{x^6\,\left (2\,x^3+1\right )} \,d x \]
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