Integrand size = 13, antiderivative size = 112 \[ \int x^6 \sqrt [3]{x+x^3} \, dx=\frac {1}{648} \sqrt [3]{x+x^3} \left (20 x-12 x^3+9 x^5+81 x^7\right )+\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{81 \sqrt {3}}+\frac {5}{243} \log \left (-x+\sqrt [3]{x+x^3}\right )-\frac {5}{486} \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.47, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2046, 2049, 2057, 335, 281, 337} \[ \int x^6 \sqrt [3]{x+x^3} \, dx=\frac {5 \left (x^2+1\right )^{2/3} x^{2/3} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{81 \sqrt {3} \left (x^3+x\right )^{2/3}}-\frac {1}{54} \sqrt [3]{x^3+x} x^3+\frac {5}{162} \sqrt [3]{x^3+x} x+\frac {1}{8} \sqrt [3]{x^3+x} x^7+\frac {1}{72} \sqrt [3]{x^3+x} x^5+\frac {5 \left (x^2+1\right )^{2/3} x^{2/3} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{162 \left (x^3+x\right )^{2/3}} \]
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Rule 281
Rule 335
Rule 337
Rule 2046
Rule 2049
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} x^7 \sqrt [3]{x+x^3}+\frac {1}{12} \int \frac {x^7}{\left (x+x^3\right )^{2/3}} \, dx \\ & = \frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {2}{27} \int \frac {x^5}{\left (x+x^3\right )^{2/3}} \, dx \\ & = -\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}+\frac {5}{81} \int \frac {x^3}{\left (x+x^3\right )^{2/3}} \, dx \\ & = \frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {10}{243} \int \frac {x}{\left (x+x^3\right )^{2/3}} \, dx \\ & = \frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {\left (10 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3}} \, dx}{243 \left (x+x^3\right )^{2/3}} \\ & = \frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {\left (10 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 )}{81 \left (x+x^3\right )^{2/3}} \\ & = \frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}-\frac {\left (5 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 )}{81 \left (x+x^3\right )^{2/3}} \\ & = \frac {5}{162} x \sqrt [3]{x+x^3}-\frac {1}{54} x^3 \sqrt [3]{x+x^3}+\frac {1}{72} x^5 \sqrt [3]{x+x^3}+\frac {1}{8} x^7 \sqrt [3]{x+x^3}+\frac {5 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 )}{81 \sqrt {3} \left (x+x^3\right )^{2/3}}+\frac {5 x^{2/3} \left (1+x^2\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{162 \left (x+x^3\right )^{2/3}} \\ \end{align*}
Time = 3.23 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.63 \[ \int x^6 \sqrt [3]{x+x^3} \, dx=\frac {\sqrt [3]{x+x^3} \left (60 x^{4/3} \sqrt [3]{1+x^2}-36 x^{10/3} \sqrt [3]{1+x^2}+27 x^{16/3} \sqrt [3]{1+x^2}+243 x^{22/3} \sqrt [3]{1+x^2}+40 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )+40 \log \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )-20 \log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )}{1944 \sqrt [3]{x} \sqrt [3]{1+x^2}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.95 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.15
method | result | size |
meijerg | \(\frac {3 x^{\frac {22}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], -x^{2}\right )}{22}\) | \(17\) |
pseudoelliptic | \(-\frac {x^{4} \left (-243 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x^{7}-27 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x^{5}+36 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x^{3}+40 \sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )-60 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +20 \ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )-40 \ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x}{x}\right )\right )}{1944 {\left ({\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}\right )}^{4} {\left ({\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x \right )}^{4}}\) | \(180\) |
trager | \(\frac {x \left (81 x^{6}+9 x^{4}-12 x^{2}+20\right ) \left (x^{3}+x \right )^{\frac {1}{3}}}{648}-\frac {5 \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 )}{243}-\frac {5 \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 )}{81}+\frac {5 \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 )}{81}\) | \(447\) |
risch | \(\text {Expression too large to display}\) | \(746\) |
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Time = 0.40 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.96 \[ \int x^6 \sqrt [3]{x+x^3} \, dx=\frac {5}{243} \, \sqrt {3} \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (539 \, x^{2} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2205 \, x^{2} + 2197}\right ) + \frac {1}{648} \, {\left (81 \, x^{7} + 9 \, x^{5} - 12 \, x^{3} + 20 \, x\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}} + \frac {5}{486} \, \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) \]
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\[ \int x^6 \sqrt [3]{x+x^3} \, dx=\int x^{6} \sqrt [3]{x \left (x^{2} + 1\right )}\, dx \]
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\[ \int x^6 \sqrt [3]{x+x^3} \, dx=\int { {\left (x^{3} + x\right )}^{\frac {1}{3}} x^{6} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.87 \[ \int x^6 \sqrt [3]{x+x^3} \, dx=\frac {1}{648} \, {\left (20 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {10}{3}} - 72 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {7}{3}} + 93 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} + 40 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )} x^{8} - \frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {5}{486} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{243} \, \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Timed out. \[ \int x^6 \sqrt [3]{x+x^3} \, dx=\int x^6\,{\left (x^3+x\right )}^{1/3} \,d x \]
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