Integrand size = 13, antiderivative size = 90 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=-\frac {\left (1+x^5\right )^{2/3}}{5 x^5}+\frac {2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^5}}{\sqrt {3}}\right )}{5 \sqrt {3}}+\frac {2}{15} \log \left (-1+\sqrt [3]{1+x^5}\right )-\frac {1}{15} \log \left (1+\sqrt [3]{1+x^5}+\left (1+x^5\right )^{2/3}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {272, 43, 57, 632, 210, 31} \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=\frac {2 \arctan \left (\frac {2 \sqrt [3]{x^5+1}+1}{\sqrt {3}}\right )}{5 \sqrt {3}}-\frac {\left (x^5+1\right )^{2/3}}{5 x^5}+\frac {1}{5} \log \left (1-\sqrt [3]{x^5+1}\right )-\frac {\log (x)}{3} \]
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Rule 31
Rule 43
Rule 57
Rule 210
Rule 272
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \text {Subst}\left (\int \frac {(1+x)^{2/3}}{x^2} \, dx,x,x^5\right ) \\ & = -\frac {\left (1+x^5\right )^{2/3}}{5 x^5}+\frac {2}{15} \text {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^5\right ) \\ & = -\frac {\left (1+x^5\right )^{2/3}}{5 x^5}-\frac {\log (x)}{3}-\frac {1}{5} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^5}\right )+\frac {1}{5} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^5}\right ) \\ & = -\frac {\left (1+x^5\right )^{2/3}}{5 x^5}-\frac {\log (x)}{3}+\frac {1}{5} \log \left (1-\sqrt [3]{1+x^5}\right )-\frac {2}{5} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^5}\right ) \\ & = -\frac {\left (1+x^5\right )^{2/3}}{5 x^5}+\frac {2 \arctan \left (\frac {1+2 \sqrt [3]{1+x^5}}{\sqrt {3}}\right )}{5 \sqrt {3}}-\frac {\log (x)}{3}+\frac {1}{5} \log \left (1-\sqrt [3]{1+x^5}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=\frac {1}{15} \left (-\frac {3 \left (1+x^5\right )^{2/3}}{x^5}+2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^5}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1+x^5}\right )-\log \left (1+\sqrt [3]{1+x^5}+\left (1+x^5\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 6.66 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {\pi \sqrt {3}\, x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], -x^{5}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+5 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{5}}\right )}{15 \pi }\) | \(76\) |
risch | \(-\frac {\left (x^{5}+1\right )^{\frac {2}{3}}}{5 x^{5}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{5} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{5}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+5 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{15 \pi }\) | \(76\) |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{5}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right ) x^{5}+2 \ln \left (-1+\left (x^{5}+1\right )^{\frac {1}{3}}\right ) x^{5}-\ln \left (1+\left (x^{5}+1\right )^{\frac {1}{3}}+\left (x^{5}+1\right )^{\frac {2}{3}}\right ) x^{5}-3 \left (x^{5}+1\right )^{\frac {2}{3}}}{15 \left (-1+\left (x^{5}+1\right )^{\frac {1}{3}}\right ) \left (1+\left (x^{5}+1\right )^{\frac {1}{3}}+\left (x^{5}+1\right )^{\frac {2}{3}}\right )}\) | \(104\) |
trager | \(-\frac {\left (x^{5}+1\right )^{\frac {2}{3}}}{5 x^{5}}-\frac {2 \ln \left (-\frac {132201 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}+214716 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}+82515 x^{5}+137820 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {2}{3}}-132201 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+137820 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {1}{3}}+124709 \left (x^{5}+1\right )^{\frac {2}{3}}+93753 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+124709 \left (x^{5}+1\right )^{\frac {1}{3}}+110020}{x^{5}}\right )}{15}-\frac {2 \ln \left (-\frac {132201 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}+214716 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}+82515 x^{5}+137820 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {2}{3}}-132201 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+137820 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {1}{3}}+124709 \left (x^{5}+1\right )^{\frac {2}{3}}+93753 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+124709 \left (x^{5}+1\right )^{\frac {1}{3}}+110020}{x^{5}}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{5}+\frac {2 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}-16 x^{5}-60 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {2}{3}}-27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-60 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {1}{3}}-37 \left (x^{5}+1\right )^{\frac {2}{3}}-69 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-37 \left (x^{5}+1\right )^{\frac {1}{3}}-40}{x^{5}}\right )}{5}\) | \(441\) |
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Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=\frac {2 \, \sqrt {3} x^{5} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{5} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - x^{5} \log \left ({\left (x^{5} + 1\right )}^{\frac {2}{3}} + {\left (x^{5} + 1\right )}^{\frac {1}{3}} + 1\right ) + 2 \, x^{5} \log \left ({\left (x^{5} + 1\right )}^{\frac {1}{3}} - 1\right ) - 3 \, {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{15 \, x^{5}} \]
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Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.38 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{5}}} \right )}}{5 x^{\frac {5}{3}} \Gamma \left (\frac {4}{3}\right )} \]
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Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=\frac {2}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{5} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{5} + 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} - \frac {1}{15} \, \log \left ({\left (x^{5} + 1\right )}^{\frac {2}{3}} + {\left (x^{5} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{15} \, \log \left ({\left (x^{5} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.74 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=\frac {2}{15} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{5} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{5} + 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} - \frac {1}{15} \, \log \left ({\left (x^{5} + 1\right )}^{\frac {2}{3}} + {\left (x^{5} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {2}{15} \, \log \left ({\left | {\left (x^{5} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Time = 5.71 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {\left (1+x^5\right )^{2/3}}{x^6} \, dx=\frac {2\,\ln \left (\frac {4\,{\left (x^5+1\right )}^{1/3}}{25}-\frac {4}{25}\right )}{15}+\ln \left (\frac {4\,{\left (x^5+1\right )}^{1/3}}{25}-9\,{\left (-\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )}^2\right )\,\left (-\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )-\ln \left (\frac {4\,{\left (x^5+1\right )}^{1/3}}{25}-9\,{\left (\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )}^2\right )\,\left (\frac {1}{15}+\frac {\sqrt {3}\,1{}\mathrm {i}}{15}\right )-\frac {{\left (x^5+1\right )}^{2/3}}{5\,x^5} \]
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