Integrand size = 24, antiderivative size = 154 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\frac {2}{3} \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {2}{3} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {1}{3} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
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Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.86, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2081, 6857, 129, 525, 524} \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\frac {4 \sqrt [4]{x^4-x^3} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x,\sqrt [3]{-1} x\right )}{9 \sqrt [4]{1-x}}+\frac {4 \sqrt [4]{x^4-x^3} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x,-(-1)^{2/3} x\right )}{9 \sqrt [4]{1-x}}+\frac {4 \sqrt [4]{x^4-x^3} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-x,x\right )}{9 \sqrt [4]{1-x}} \]
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Rule 129
Rule 524
Rule 525
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x} \left (1+x^3\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}} \\ & = \frac {\sqrt [4]{-x^3+x^4} \int \left (-\frac {\sqrt [4]{-1+x}}{3 (-1-x) \sqrt [4]{x}}-\frac {\sqrt [4]{-1+x}}{3 \sqrt [4]{x} \left (-1+\sqrt [3]{-1} x\right )}-\frac {\sqrt [4]{-1+x}}{3 \sqrt [4]{x} \left (-1-(-1)^{2/3} x\right )}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}} \\ & = -\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x}}{(-1-x) \sqrt [4]{x}} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x} \left (-1+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x} \left (-1-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [4]{-1+x} x^{3/4}} \\ & = -\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{-1-x^4} \, dx,x,\sqrt [4]{x}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{-1+\sqrt [3]{-1} x^4} \, dx,x,\sqrt [4]{x}\right )}{3 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{-1+x^4}}{-1-(-1)^{2/3} x^4} \, dx,x,\sqrt [4]{x}\right )}{3 \sqrt [4]{-1+x} x^{3/4}} \\ & = -\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{-1-x^4} \, dx,x,\sqrt [4]{x}\right )}{3 \sqrt [4]{1-x} x^{3/4}}-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{-1+\sqrt [3]{-1} x^4} \, dx,x,\sqrt [4]{x}\right )}{3 \sqrt [4]{1-x} x^{3/4}}-\frac {\left (4 \sqrt [4]{-x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt [4]{1-x^4}}{-1-(-1)^{2/3} x^4} \, dx,x,\sqrt [4]{x}\right )}{3 \sqrt [4]{1-x} x^{3/4}} \\ & = \frac {4 \sqrt [4]{-x^3+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x,\sqrt [3]{-1} x\right )}{9 \sqrt [4]{1-x}}+\frac {4 \sqrt [4]{-x^3+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x,-(-1)^{2/3} x\right )}{9 \sqrt [4]{1-x}}+\frac {4 \sqrt [4]{-x^3+x^4} \operatorname {AppellF1}\left (\frac {3}{4},1,-\frac {1}{4},\frac {7}{4},-x,x\right )}{9 \sqrt [4]{1-x}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\frac {(-1+x)^{3/4} x^{9/4} \left (8 \sqrt [4]{2} \left (\arctan \left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{2}}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )-\text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-4 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{-1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]\right )}{12 \left ((-1+x) x^3\right )^{3/4}} \]
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Time = 70.90 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{3}-\frac {2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-1\right )}\right )}{3}\) | \(122\) |
trager | \(\text {Expression too large to display}\) | \(16803\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 586, normalized size of antiderivative = 3.81 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\frac {1}{6} \, \sqrt {2} \sqrt {-\sqrt {2 i \, \sqrt {3} + 2}} \log \left (\frac {\sqrt {2} {\left (i \, \sqrt {3} x + x\right )} \sqrt {-\sqrt {2 i \, \sqrt {3} + 2}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{6} \, \sqrt {2} \sqrt {-\sqrt {2 i \, \sqrt {3} + 2}} \log \left (\frac {\sqrt {2} {\left (-i \, \sqrt {3} x - x\right )} \sqrt {-\sqrt {2 i \, \sqrt {3} + 2}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{6} \, \sqrt {2} \sqrt {-\sqrt {-2 i \, \sqrt {3} + 2}} \log \left (\frac {\sqrt {2} {\left (i \, \sqrt {3} x - x\right )} \sqrt {-\sqrt {-2 i \, \sqrt {3} + 2}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{6} \, \sqrt {2} \sqrt {-\sqrt {-2 i \, \sqrt {3} + 2}} \log \left (\frac {\sqrt {2} {\left (-i \, \sqrt {3} x + x\right )} \sqrt {-\sqrt {-2 i \, \sqrt {3} + 2}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{6} \, \sqrt {2} {\left (2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (i \, \sqrt {3} x + x\right )} {\left (2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{6} \, \sqrt {2} {\left (2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (-i \, \sqrt {3} x - x\right )} {\left (2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{6} \, \sqrt {2} {\left (-2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (i \, \sqrt {3} x - x\right )} {\left (-2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{6} \, \sqrt {2} {\left (-2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (-i \, \sqrt {3} x + x\right )} {\left (-2 i \, \sqrt {3} + 2\right )}^{\frac {1}{4}} + 4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{3} \cdot 2^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{3} \cdot 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{3} i \cdot 2^{\frac {1}{4}} \log \left (\frac {i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{3} i \cdot 2^{\frac {1}{4}} \log \left (\frac {-i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
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Not integrable
Time = 0.68 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )}}{x \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{{\left (x^{3} + 1\right )} x} \,d x } \]
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.35 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.55 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\frac {1}{6} \, {\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}{6} \, {\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}{6} \, {\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}{6} \, {\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}{12} \, {\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}{12} \, {\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}{12} \, {\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}{12} \, {\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 {2}{3} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{3} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{3} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.16 \[ \int \frac {\sqrt [4]{-x^3+x^4}}{x \left (1+x^3\right )} \, dx=\int \frac {{\left (x^4-x^3\right )}^{1/4}}{x\,\left (x^3+1\right )} \,d x \]
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