Integrand size = 15, antiderivative size = 63 \[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\frac {1}{24} \sqrt [4]{-x+x^4} \left (-x^2+4 x^5\right )+\frac {1}{16} \arctan \left (\frac {x}{\sqrt [4]{-x+x^4}}\right )-\frac {1}{16} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(127\) vs. \(2(63)=126\).
Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.02, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2046, 2049, 2057, 335, 281, 338, 304, 209, 212} \[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\frac {\left (x^3-1\right )^{3/4} x^{3/4} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{16 \left (x^4-x\right )^{3/4}}-\frac {\left (x^3-1\right )^{3/4} x^{3/4} \text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{16 \left (x^4-x\right )^{3/4}}+\frac {1}{6} \sqrt [4]{x^4-x} x^5-\frac {1}{24} \sqrt [4]{x^4-x} x^2 \]
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Rule 209
Rule 212
Rule 281
Rule 304
Rule 335
Rule 338
Rule 2046
Rule 2049
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {1}{8} \int \frac {x^5}{\left (-x+x^4\right )^{3/4}} \, dx \\ & = -\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {3}{32} \int \frac {x^2}{\left (-x+x^4\right )^{3/4}} \, dx \\ & = -\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (3 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \int \frac {x^{5/4}}{\left (-1+x^3\right )^{3/4}} \, dx}{32 \left (-x+x^4\right )^{3/4}} \\ & = -\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (3 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^8}{\left (-1+x^{12}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{8 \left (-x+x^4\right )^{3/4}} \\ & = -\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,x^{3/4}\right )}{8 \left (-x+x^4\right )^{3/4}} \\ & = -\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{8 \left (-x+x^4\right )^{3/4}} \\ & = -\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{16 \left (-x+x^4\right )^{3/4}}+\frac {\left (x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{16 \left (-x+x^4\right )^{3/4}} \\ & = -\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}+\frac {x^{3/4} \left (-1+x^3\right )^{3/4} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{16 \left (-x+x^4\right )^{3/4}}-\frac {x^{3/4} \left (-1+x^3\right )^{3/4} \text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{16 \left (-x+x^4\right )^{3/4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\frac {x^2 \sqrt [4]{x \left (-1+x^3\right )} \left (-\left (1-x^3\right )^{5/4}+\operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{4},\frac {7}{4},x^3\right )\right )}{6 \sqrt [4]{1-x^3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.52
method | result | size |
meijerg | \(\frac {4 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {21}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], x^{3}\right )}{21 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}}}\) | \(33\) |
pseudoelliptic | \(\frac {x^{2} \left (16 \left (x^{4}-x \right )^{\frac {1}{4}} x^{5}-4 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}-6 \arctan \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}}{x}\right )+3 \ln \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}-x}{x}\right )-3 \ln \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}+x}{x}\right )\right )}{96 {\left (\left (x^{4}-x \right )^{\frac {1}{4}}-x \right )}^{2} \left (x^{2}+\sqrt {x^{4}-x}\right )^{2} {\left (\left (x^{4}-x \right )^{\frac {1}{4}}+x \right )}^{2}}\) | \(132\) |
trager | \(\frac {x^{2} \left (4 x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{24}+\frac {\ln \left (-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x \sqrt {x^{4}-x}-2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+2 x^{3}-1\right )}{32}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \sqrt {x^{4}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{32}\) | \(140\) |
risch | \(\frac {x^{2} \left (4 x^{3}-1\right ) {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}}}{24}+\frac {\left (-\frac {\ln \left (\frac {2 x^{9}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}-5 x^{6}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}-4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}-1}{\left (x^{2}+x +1\right )^{2} \left (x -1\right )^{2}}\right )}{32}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 x^{9}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}-5 x^{6}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}-1}{\left (x^{2}+x +1\right )^{2} \left (x -1\right )^{2}}\right )}{32}\right ) {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (x^{3} \left (x^{3}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{3}-1\right )}\) | \(450\) |
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Time = 1.18 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.57 \[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\frac {1}{24} \, {\left (4 \, x^{5} - x^{2}\right )} {\left (x^{4} - x\right )}^{\frac {1}{4}} - \frac {1}{32} \, \arctan \left (2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}}\right ) + \frac {1}{32} \, \log \left (2 \, x^{3} - 2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} - x} x - 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1\right ) \]
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\[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\int x^{4} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
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\[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\int { {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{4} \,d x } \]
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Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.08 \[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\frac {1}{24} \, {\left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right )} x^{6} - \frac {1}{16} \, \arctan \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{32} \, \log \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{32} \, \log \left ({\left | {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
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Timed out. \[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\int x^4\,{\left (x^4-x\right )}^{1/4} \,d x \]
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