Integrand size = 9, antiderivative size = 94 \[ \int \sqrt [3]{x+x^3} \, dx=\frac {1}{2} x \sqrt [3]{x+x^3}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x+\sqrt [3]{x+x^3}\right )+\frac {1}{12} \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2029, 2057, 335, 281, 337} \[ \int \sqrt [3]{x+x^3} \, dx=-\frac {\left (x^2+1\right )^{2/3} x^{2/3} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} \left (x^3+x\right )^{2/3}}+\frac {1}{2} \sqrt [3]{x^3+x} x-\frac {\left (x^2+1\right )^{2/3} x^{2/3} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 \left (x^3+x\right )^{2/3}} \]
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Rule 281
Rule 335
Rule 337
Rule 2029
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x \sqrt [3]{x+x^3}+\frac {1}{3} \int \frac {x}{\left (x+x^3\right )^{2/3}} \, dx \\ & = \frac {1}{2} x \sqrt [3]{x+x^3}+\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}{3 \left (x+x^3\right )^{2/3}} \\ & = \frac {1}{2} x \sqrt [3]{x+x^3}+\frac {\left (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 {1}{2} x \sqrt [3]{x+x^3}+\frac {\left (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 {1}{2} x \sqrt [3]{x+x^3}-\frac {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 \sqrt {3} \left (x+x^3\right )^{2/3}}-\frac {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 = 0.53 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.41 \[ \int \sqrt [3]{x+x^3} \, dx=\frac {\sqrt [3]{x+x^3} \left (6 x^{4/3} \sqrt [3]{1+x^2}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )-2 \log \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )+\log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )}{12 \sqrt [3]{x} \sqrt [3]{1+x^2}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 6.50 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.18
method | result | size |
meijerg | \(\frac {3 x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{2}\right )}{4}\) | \(17\) |
pseudoelliptic | \(\frac {x \left (6 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}} x +2 \sqrt {3}\, \arctan \left (\frac {\left (2 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )-2 \ln \left (\frac {{\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}-x}{x}\right )+\ln \left (\frac {{\left (\left (x^{2}+1\right ) x \right )}^{\frac {2}{3}}+{\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )\right )}{12 \left ({\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}-x \right ) \left ({\left (\left (x^{2}+1\right ) x \right )}^{\frac {2}{3}}+{\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}} x +x^{2}\right )}\) | \(134\) |
trager | \(\frac {x \left (x^{3}+x \right )^{\frac {1}{3}}}{2}-\frac {\ln \left (-36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}-27 \left (x^{3}+x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -33 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-9 \left (x^{3}+x \right )^{\frac {2}{3}}-15 x \left (x^{3}+x \right )^{\frac {1}{3}}+20 x^{2}-51 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+15\right )}{6}+\frac {\operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-72 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x \right )^{\frac {2}{3}}+45 \left (x^{3}+x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +30 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+15 \left (x^{3}+x \right )^{\frac {2}{3}}+9 x \left (x^{3}+x \right )^{\frac {1}{3}}-25 x^{2}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-10\right )}{2}\) | \(287\) |
risch | \(\frac {{\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}} x}{2}+\frac {\left (-\frac {\ln \left (-\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}-38 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}+18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+16 x^{4}+30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-70 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-96 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+18 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}+28 x^{2}+60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}-32 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )+12}{x^{2}+1}\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}+20 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-48 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-100 x^{4}+30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+14 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+36 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}-48 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}-140 x^{2}+60 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-40}{x^{2}+1}\right )}{12}\right ) {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}} {\left (\left (x^{2}+1\right )^{2} x^{2}\right )}^{\frac {1}{3}}}{\left (x^{2}+1\right ) x}\) | \(496\) |
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Time = 0.35 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.96 \[ \int \sqrt [3]{x+x^3} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (539 \, x^{2} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2205 \, x^{2} + 2197}\right ) + \frac {1}{2} \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \frac {1}{12} \, \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) \]
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\[ \int \sqrt [3]{x+x^3} \, dx=\int \sqrt [3]{x^{3} + x}\, dx \]
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\[ \int \sqrt [3]{x+x^3} \, dx=\int { {\left (x^{3} + x\right )}^{\frac {1}{3}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.71 \[ \int \sqrt [3]{x+x^3} \, dx=\frac {1}{2} \, x^{2} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{12} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Time = 5.84 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.29 \[ \int \sqrt [3]{x+x^3} \, dx=\frac {3\,x\,{\left (x^3+x\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {2}{3};\ \frac {5}{3};\ -x^2\right )}{4\,{\left (x^2+1\right )}^{1/3}} \]
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