Integrand size = 13, antiderivative size = 96 \[ \int \frac {\sqrt [3]{x+x^3}}{x^2} \, dx=-\frac {3 \sqrt [3]{x+x^3}}{2 x}-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )-\frac {1}{2} \log \left (-x+\sqrt [3]{x+x^3}\right )+\frac {1}{4} \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2045, 2057, 335, 281, 337} \[ \int \frac {\sqrt [3]{x+x^3}}{x^2} \, dx=-\frac {\sqrt {3} x^{2/3} \left (x^2+1\right )^{2/3} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \left (x^3+x\right )^{2/3}}-\frac {3 \sqrt [3]{x^3+x}}{2 x}-\frac {3 x^{2/3} \left (x^2+1\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 \left (x^3+x\right )^{2/3}} \]
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Rule 281
Rule 335
Rule 337
Rule 2045
Rule 2057
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \sqrt [3]{x+x^3}}{2 x}+\int \frac {x}{\left (x+x^3\right )^{2/3}} \, dx \\ & = -\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (x^{2/3} \left (1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3}} \, dx}{\left (x+x^3\right )^{2/3}} \\ & = -\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (x+x^3\right )^{2/3}} \\ & = -\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 \left (x+x^3\right )^{2/3}} \\ & = -\frac {3 \sqrt [3]{x+x^3}}{2 x}-\frac {\sqrt {3} x^{2/3} \left (1+x^2\right )^{2/3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \left (x+x^3\right )^{2/3}}-\frac {3 x^{2/3} \left (1+x^2\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 \left (x+x^3\right )^{2/3}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt [3]{x+x^3}}{x^2} \, dx=\frac {\sqrt [3]{x+x^3} \left (-6 \sqrt [3]{1+x^2}-2 \sqrt {3} x^{2/3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )-2 x^{2/3} \log \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )+x^{2/3} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )}{4 x \sqrt [3]{1+x^2}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 4.96 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.18
method | result | size |
meijerg | \(-\frac {3 \operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], -x^{2}\right )}{2 x^{\frac {2}{3}}}\) | \(17\) |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right ) x -2 \ln \left (\frac {{\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}-x}{x}\right ) x +\ln \left (\frac {{\left (\left (x^{2}+1\right ) x \right )}^{\frac {2}{3}}+{\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right ) x -6 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}}{4 x}\) | \(98\) |
trager | \(-\frac {3 \left (x^{3}+x \right )^{\frac {1}{3}}}{2 x}+\frac {\ln \left (45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-72 \left (x^{3}+x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -87 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+15 x \left (x^{3}+x \right )^{\frac {1}{3}}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+20 x^{2}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+8\right )}{2}-\frac {3 \ln \left (45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-72 \left (x^{3}+x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -87 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+15 x \left (x^{3}+x \right )^{\frac {1}{3}}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+20 x^{2}-18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+8\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{2}+\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}+72 \left (x^{3}+x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-9 \left (x^{3}+x \right )^{\frac {2}{3}}-9 x \left (x^{3}+x \right )^{\frac {1}{3}}-45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-4 x^{2}+48 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-3\right )}{2}\) | \(432\) |
risch | \(\text {Expression too large to display}\) | \(731\) |
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Time = 0.34 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt [3]{x+x^3}}{x^2} \, dx=-\frac {2 \, \sqrt {3} x \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (539 \, x^{2} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2205 \, x^{2} + 2197}\right ) + x \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) + 6 \, {\left (x^{3} + x\right )}^{\frac {1}{3}}}{4 \, x} \]
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\[ \int \frac {\sqrt [3]{x+x^3}}{x^2} \, dx=\int \frac {\sqrt [3]{x \left (x^{2} + 1\right )}}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt [3]{x+x^3}}{x^2} \, dx=\int { \frac {{\left (x^{3} + x\right )}^{\frac {1}{3}}}{x^{2}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt [3]{x+x^3}}{x^2} \, dx=\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {3}{2} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + \frac {1}{4} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{2} \, \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {\sqrt [3]{x+x^3}}{x^2} \, dx=\int \frac {{\left (x^3+x\right )}^{1/3}}{x^2} \,d x \]
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