Integrand size = 27, antiderivative size = 147 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\sqrt [4]{-x^3+x^4}-\frac {7}{2} \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {7}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
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Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 695, normalized size of antiderivative = 4.73, number of steps used = 25, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {2081, 6860, 103, 163, 65, 246, 218, 212, 209, 95, 304, 211, 214} \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {\left (-\frac {-\sqrt {3}+i}{\sqrt {3}+i}\right )^{3/4} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \arctan \left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}-\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}-\frac {\left (-\frac {-\sqrt {3}+i}{\sqrt {3}+i}\right )^{3/4} \left (\sqrt {3}+i\right ) \sqrt [4]{x^4-x^3} \text {arctanh}\left (\frac {\sqrt [4]{-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{x-1} x^{3/4}}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{x^4-x^3}+\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{x^4-x^3} \]
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Rule 65
Rule 95
Rule 103
Rule 163
Rule 209
Rule 211
Rule 212
Rule 214
Rule 218
Rule 246
Rule 304
Rule 2081
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} (1+x)}{1-x+x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {\sqrt [4]{-x^3+x^4} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{-1+x} x^{3/4}}{-1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{-1+x} x^{3/4}}{-1+i \sqrt {3}+2 x}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-1-i \sqrt {3}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-1+i \sqrt {3}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{4} \left (1+i \sqrt {3}\right )+\frac {1}{2} \left (-1-2 i \sqrt {3}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1-i \sqrt {3}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{4} \left (1-i \sqrt {3}\right )+\frac {1}{2} \left (-1+2 i \sqrt {3}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1+i \sqrt {3}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1-i \sqrt {3}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1+i \sqrt {3}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{8 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{8 \sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-\frac {\left (4 \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-1-i \sqrt {3}-\left (1-i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{-1+i \sqrt {3}-\left (1+i \sqrt {3}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{2 \sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 i \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}-\sqrt {-i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {-i+\sqrt {3}} \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 i \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {i+\sqrt {3}}+\sqrt {-i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {-i+\sqrt {3}} \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 i \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-i+\sqrt {3}}-\sqrt {i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {i+\sqrt {3}} \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 i \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-i+\sqrt {3}}+\sqrt {i+\sqrt {3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt {i+\sqrt {3}} \sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {2 \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x}}\right )}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x}}\right )}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-1-2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-1+2 i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (1+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x}}\right )}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x} x^{3/4}}+\frac {2 \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-x^3+x^4} \arctan \left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (7-i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (7+i \sqrt {3}\right ) \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x}}\right )}{\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-1+x} x^{3/4}}-\frac {2 \sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{-x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.08 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\frac {(-1+x)^{3/4} x^{9/4} \left (4 \sqrt [4]{-1+x} x^{3/4}-14 \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+14 \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+8 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{4 \left ((-1+x) x^3\right )^{3/4}} \]
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Time = 31.79 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}+\frac {7 \ln \left (\frac {x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{4}-\frac {7 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )}{4}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-1\right )}\right )+\frac {7 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{2}\) | \(116\) |
trager | \(\text {Expression too large to display}\) | \(2951\) |
risch | \(\text {Expression too large to display}\) | \(3739\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.27 (sec) , antiderivative size = 485, normalized size of antiderivative = 3.30 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {-\sqrt {2 \, \sqrt {-3} + 2}} \log \left (\frac {\sqrt {2} x \sqrt {-\sqrt {2 \, \sqrt {-3} + 2}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {-\sqrt {2 \, \sqrt {-3} + 2}} \log \left (-\frac {\sqrt {2} x \sqrt {-\sqrt {2 \, \sqrt {-3} + 2}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {-\sqrt {-2 \, \sqrt {-3} + 2}} \log \left (\frac {\sqrt {2} x \sqrt {-\sqrt {-2 \, \sqrt {-3} + 2}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {-\sqrt {-2 \, \sqrt {-3} + 2}} \log \left (-\frac {\sqrt {2} x \sqrt {-\sqrt {-2 \, \sqrt {-3} + 2}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} {\left (2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} x {\left (2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} {\left (2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} x {\left (2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} {\left (-2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} x {\left (-2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} {\left (-2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} x {\left (-2 \, \sqrt {-3} + 2\right )}^{\frac {1}{4}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} + \frac {7}{2} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {7}{4} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {7}{4} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
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Not integrable
Time = 1.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.14 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x + 1\right )}{x^{2} - x + 1}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.18 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x + 1\right )}}{x^{2} - x + 1} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.32 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.61 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=-\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} - \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} - \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} + \sqrt {2} + 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} + \sqrt {2} - 4 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{4} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {1}{4} \, {\left (\sqrt {6} + \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} + \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) - \frac {1}{4} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + \frac {1}{4} \, {\left (\sqrt {6} - \sqrt {2}\right )} \log \left (-\frac {1}{2} \, {\left (\sqrt {6} - \sqrt {2}\right )} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \sqrt {-\frac {1}{x} + 1} + 1\right ) + x {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + \frac {7}{2} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {7}{4} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {7}{4} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.18 \[ \int \frac {(1+x) \sqrt [4]{-x^3+x^4}}{1-x+x^2} \, dx=\int \frac {{\left (x^4-x^3\right )}^{1/4}\,\left (x+1\right )}{x^2-x+1} \,d x \]
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