Integrand size = 15, antiderivative size = 100 \[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=-\frac {3 \sqrt [3]{x^2+x^3}}{x}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.47, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2045, 2057, 61} \[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=-\frac {\sqrt {3} (x+1)^{2/3} x^{4/3} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x+1}}+\frac {1}{\sqrt {3}}\right )}{\left (x^3+x^2\right )^{2/3}}-\frac {3 \sqrt [3]{x^3+x^2}}{x}-\frac {(x+1)^{2/3} x^{4/3} \log (x+1)}{2 \left (x^3+x^2\right )^{2/3}}-\frac {3 (x+1)^{2/3} x^{4/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x+1}}-1\right )}{2 \left (x^3+x^2\right )^{2/3}} \]
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Rule 61
Rule 2045
Rule 2057
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \sqrt [3]{x^2+x^3}}{x}+\int \frac {x}{\left (x^2+x^3\right )^{2/3}} \, dx \\ & = -\frac {3 \sqrt [3]{x^2+x^3}}{x}+\frac {\left (x^{4/3} (1+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (1+x)^{2/3}} \, dx}{\left (x^2+x^3\right )^{2/3}} \\ & = -\frac {3 \sqrt [3]{x^2+x^3}}{x}-\frac {\sqrt {3} x^{4/3} (1+x)^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{\left (x^2+x^3\right )^{2/3}}-\frac {x^{4/3} (1+x)^{2/3} \log (1+x)}{2 \left (x^2+x^3\right )^{2/3}}-\frac {3 x^{4/3} (1+x)^{2/3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{1+x}}\right )}{2 \left (x^2+x^3\right )^{2/3}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=-\frac {x (1+x)^{2/3} \left (6 \sqrt [3]{1+x}+2 \sqrt {3} \sqrt [3]{x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x}}\right )+2 \sqrt [3]{x} \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x}\right )-\sqrt [3]{x} \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )\right )}{2 \left (x^2 (1+x)\right )^{2/3}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 0.70 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.15
method | result | size |
meijerg | \(-\frac {3 \operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], -x \right )}{x^{\frac {1}{3}}}\) | \(15\) |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right ) x +\ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right ) x -2 \ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}-x}{x}\right ) x -6 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{2 x}\) | \(98\) |
risch | \(-\frac {3 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}+\frac {\left (-\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-48 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}+30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+14 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +36 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}-96 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x +64 x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-96 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}+112 x +48}{1+x}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+24 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -19 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}-28 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -30 \left (x^{3}+2 x^{2}+x \right )^{\frac {2}{3}}+48 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}} x -10 x^{2}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )+48 \left (x^{3}+2 x^{2}+x \right )^{\frac {1}{3}}-14 x -4}{1+x}\right )}{2}\right ) \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} \left (\left (1+x \right )^{2} x \right )^{\frac {1}{3}}}{x \left (1+x \right )}\) | \(450\) |
trager | \(-\frac {3 \left (x^{3}+x^{2}\right )^{\frac {1}{3}}}{x}-3 \ln \left (-\frac {36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-45 \left (x^{3}+x^{2}\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}-27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -9 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-5 x^{2}-2 x}{x}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-\frac {36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x +45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}+45 \left (x^{3}+x^{2}\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}+51 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -24 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-24 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-20 x^{2}-15 x}{x}\right )+\ln \left (-\frac {36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-36 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x -45 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-45 \left (x^{3}+x^{2}\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}-27 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -9 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}-9 x \left (x^{3}+x^{2}\right )^{\frac {1}{3}}-5 x^{2}-2 x}{x}\right )\) | \(483\) |
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Time = 0.24 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) - 2 \, x \log \left (-\frac {x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + x \log \left (\frac {x^{2} + {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 6 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{2 \, x} \]
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\[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=\int \frac {\sqrt [3]{x^{2} \left (x + 1\right )}}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=\int { \frac {{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x^{2}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - 3 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + \frac {1}{2} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Time = 6.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt [3]{x^2+x^3}}{x^2} \, dx=-\frac {3\,{\left (x^2\,\left (x+1\right )\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},-\frac {1}{3};\ \frac {2}{3};\ -x\right )}{x\,{\left (x+1\right )}^{1/3}} \]
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