Integrand size = 22, antiderivative size = 123 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=-\arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {1}{2} \text {RootSum}\left [1-5 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-5 \text {$\#$1}^3+4 \text {$\#$1}^7}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(378\) vs. \(2(123)=246\).
Time = 0.47 (sec) , antiderivative size = 378, normalized size of antiderivative = 3.07, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {2081, 919, 65, 338, 304, 209, 212, 6860, 95} \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=-\frac {\sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{\frac {1}{2} \left (23-\sqrt {17}\right )} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{\frac {1}{2} \left (23+\sqrt {17}\right )} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{5+\sqrt {17}} \sqrt [4]{x}}{\sqrt {2} \sqrt [4]{x+1}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{\frac {1}{2} \left (23-\sqrt {17}\right )} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{\frac {1}{2} \left (23+\sqrt {17}\right )} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{5+\sqrt {17}} \sqrt [4]{x}}{\sqrt {2} \sqrt [4]{x+1}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{x+1}} \]
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Rule 65
Rule 95
Rule 209
Rule 212
Rule 304
Rule 338
Rule 919
Rule 2081
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{x^3+x^4} \int \frac {x^{3/4} \sqrt [4]{1+x}}{-2+x+2 x^2} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\sqrt [4]{x^3+x^4} \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \int \frac {2+x}{\sqrt [4]{x} (1+x)^{3/4} \left (-2+x+2 x^2\right )} \, dx}{2 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\sqrt [4]{x^3+x^4} \int \left (\frac {1+\frac {7}{\sqrt {17}}}{\sqrt [4]{x} (1+x)^{3/4} \left (1-\sqrt {17}+4 x\right )}+\frac {1-\frac {7}{\sqrt {17}}}{\sqrt [4]{x} (1+x)^{3/4} \left (1+\sqrt {17}+4 x\right )}\right ) \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (17-7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (1+\sqrt {17}+4 x\right )} \, dx}{34 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (17+7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (1-\sqrt {17}+4 x\right )} \, dx}{34 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (17-7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1+\sqrt {17}-\left (-3+\sqrt {17}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (17+7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-\sqrt {17}-\left (-3-\sqrt {17}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\sqrt {2} \left (17-7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {5+\sqrt {17}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 \left (3-\sqrt {17}\right ) x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\sqrt {2} \left (17-7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {5+\sqrt {17}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 \left (3-\sqrt {17}\right ) x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\sqrt {2} \left (17+7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {5-\sqrt {17}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 \left (3+\sqrt {17}\right ) x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\sqrt {2} \left (17+7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {5-\sqrt {17}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 \left (3+\sqrt {17}\right ) x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{\frac {1}{2} \left (23-\sqrt {17}\right )} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{\frac {1}{2} \left (23+\sqrt {17}\right )} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{5+\sqrt {17}} \sqrt [4]{x}}{\sqrt {2} \sqrt [4]{1+x}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{\frac {1}{2} \left (23-\sqrt {17}\right )} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{\frac {1}{2} \left (23+\sqrt {17}\right )} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{5+\sqrt {17}} \sqrt [4]{x}}{\sqrt {2} \sqrt [4]{1+x}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{1+x}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=-\frac {x^{9/4} (1+x)^{3/4} \left (8 \arctan \left (\sqrt [4]{\frac {x}{1+x}}\right )-8 \text {arctanh}\left (\sqrt [4]{\frac {x}{1+x}}\right )+\text {RootSum}\left [1-5 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log (x)+4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-8 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-5 \text {$\#$1}^3+4 \text {$\#$1}^7}\&\right ]\right )}{8 \left (x^3 (1+x)\right )^{3/4}} \]
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Time = 11.78 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-5 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{4}-1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (4 \textit {\_R}^{4}-5\right )}\right )}{2}+\frac {\ln \left (\frac {x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{2}-\frac {\ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )}{2}+\arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )\) | \(111\) |
trager | \(\text {Expression too large to display}\) | \(3016\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 570, normalized size of antiderivative = 4.63 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=-\frac {1}{68} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} \log \left (\frac {\sqrt {34} {\left (\sqrt {17} x + 17 \, x\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} + 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{68} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} \log \left (-\frac {\sqrt {34} {\left (\sqrt {17} x + 17 \, x\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} - 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{68} \, \sqrt {34} \sqrt {-\sqrt {2} \sqrt {\sqrt {17} + 23}} \log \left (\frac {\sqrt {34} {\left (\sqrt {17} x + 17 \, x\right )} \sqrt {-\sqrt {2} \sqrt {\sqrt {17} + 23}} + 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{68} \, \sqrt {34} \sqrt {-\sqrt {2} \sqrt {\sqrt {17} + 23}} \log \left (-\frac {\sqrt {34} {\left (\sqrt {17} x + 17 \, x\right )} \sqrt {-\sqrt {2} \sqrt {\sqrt {17} + 23}} - 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{68} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} \log \left (\frac {\sqrt {34} {\left (\sqrt {17} x - 17 \, x\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} + 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{68} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} \log \left (-\frac {\sqrt {34} {\left (\sqrt {17} x - 17 \, x\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} - 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{68} \, \sqrt {34} \sqrt {-\sqrt {2} \sqrt {-\sqrt {17} + 23}} \log \left (\frac {\sqrt {34} {\left (\sqrt {17} x - 17 \, x\right )} \sqrt {-\sqrt {2} \sqrt {-\sqrt {17} + 23}} + 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{68} \, \sqrt {34} \sqrt {-\sqrt {2} \sqrt {-\sqrt {17} + 23}} \log \left (-\frac {\sqrt {34} {\left (\sqrt {17} x - 17 \, x\right )} \sqrt {-\sqrt {2} \sqrt {-\sqrt {17} + 23}} - 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
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Not integrable
Time = 0.39 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.15 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x + 1\right )}}{2 x^{2} + x - 2}\, dx \]
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Not integrable
Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{2 \, x^{2} + x - 2} \,d x } \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{2 \, x^{2} + x - 2} \,d x } \]
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Not integrable
Time = 5.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=\int \frac {{\left (x^4+x^3\right )}^{1/4}}{2\,x^2+x-2} \,d x \]
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