Integrand size = 22, antiderivative size = 101 \[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=\frac {\sqrt [4]{-x^3+x^4} \left (327680-65536 x-262144 x^2-21945 x^3-12540 x^4-9120 x^5-7296 x^6-6144 x^7+122880 x^8\right )}{737280 x^3}+\frac {1463 \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{32768}-\frac {1463 \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{32768} \]
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Leaf count is larger than twice the leaf count of optimal. \(242\) vs. \(2(101)=202\).
Time = 0.24 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.40, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2077, 2041, 2039, 2046, 2049, 2057, 65, 246, 218, 212, 209} \[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=-\frac {1463 (x-1)^{3/4} x^{9/4} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{32768 \left (x^4-x^3\right )^{3/4}}-\frac {1463 (x-1)^{3/4} x^{9/4} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{32768 \left (x^4-x^3\right )^{3/4}}-\frac {1}{120} \sqrt [4]{x^4-x^3} x^4-\frac {19 \sqrt [4]{x^4-x^3} x^3}{1920}-\frac {209 \sqrt [4]{x^4-x^3} x}{12288}-\frac {1463 \sqrt [4]{x^4-x^3}}{49152}-\frac {4 \left (x^4-x^3\right )^{5/4}}{9 x^6}+\frac {1}{6} \sqrt [4]{x^4-x^3} x^5-\frac {16 \left (x^4-x^3\right )^{5/4}}{45 x^5}-\frac {19 \sqrt [4]{x^4-x^3} x^2}{1536} \]
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Rule 65
Rule 209
Rule 212
Rule 218
Rule 246
Rule 2039
Rule 2041
Rule 2046
Rule 2049
Rule 2057
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt [4]{-x^3+x^4}}{x^4}+x^4 \sqrt [4]{-x^3+x^4}\right ) \, dx \\ & = -\int \frac {\sqrt [4]{-x^3+x^4}}{x^4} \, dx+\int x^4 \sqrt [4]{-x^3+x^4} \, dx \\ & = \frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {1}{24} \int \frac {x^7}{\left (-x^3+x^4\right )^{3/4}} \, dx-\frac {4}{9} \int \frac {\sqrt [4]{-x^3+x^4}}{x^3} \, dx \\ & = -\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {19}{480} \int \frac {x^6}{\left (-x^3+x^4\right )^{3/4}} \, dx \\ & = -\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {19}{512} \int \frac {x^5}{\left (-x^3+x^4\right )^{3/4}} \, dx \\ & = -\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {209 \int \frac {x^4}{\left (-x^3+x^4\right )^{3/4}} \, dx}{6144} \\ & = -\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {1463 \int \frac {x^3}{\left (-x^3+x^4\right )^{3/4}} \, dx}{49152} \\ & = -\frac {1463 \sqrt [4]{-x^3+x^4}}{49152}-\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {1463 \int \frac {x^2}{\left (-x^3+x^4\right )^{3/4}} \, dx}{65536} \\ & = -\frac {1463 \sqrt [4]{-x^3+x^4}}{49152}-\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {\left (1463 (-1+x)^{3/4} x^{9/4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{65536 \left (-x^3+x^4\right )^{3/4}} \\ & = -\frac {1463 \sqrt [4]{-x^3+x^4}}{49152}-\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {\left (1463 (-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{16384 \left (-x^3+x^4\right )^{3/4}} \\ & = -\frac {1463 \sqrt [4]{-x^3+x^4}}{49152}-\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {\left (1463 (-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{16384 \left (-x^3+x^4\right )^{3/4}} \\ & = -\frac {1463 \sqrt [4]{-x^3+x^4}}{49152}-\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {\left (1463 (-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32768 \left (-x^3+x^4\right )^{3/4}}-\frac {\left (1463 (-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32768 \left (-x^3+x^4\right )^{3/4}} \\ & = -\frac {1463 \sqrt [4]{-x^3+x^4}}{49152}-\frac {209 x \sqrt [4]{-x^3+x^4}}{12288}-\frac {19 x^2 \sqrt [4]{-x^3+x^4}}{1536}-\frac {19 x^3 \sqrt [4]{-x^3+x^4}}{1920}-\frac {1}{120} x^4 \sqrt [4]{-x^3+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x^3+x^4}-\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{45 x^5}-\frac {1463 (-1+x)^{3/4} x^{9/4} \arctan \left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32768 \left (-x^3+x^4\right )^{3/4}}-\frac {1463 (-1+x)^{3/4} x^{9/4} \text {arctanh}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{32768 \left (-x^3+x^4\right )^{3/4}} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=\frac {(-1+x)^{3/4} \left (2 \sqrt [4]{-1+x} \left (327680-65536 x-262144 x^2-21945 x^3-12540 x^4-9120 x^5-7296 x^6-6144 x^7+122880 x^8\right )+65835 x^{9/4} \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-65835 x^{9/4} \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{1474560 \left ((-1+x) x^3\right )^{3/4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.70 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.63
method | result | size |
meijerg | \(\frac {4 \operatorname {signum}\left (-1+x \right )^{\frac {1}{4}} x^{\frac {23}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {23}{4}\right ], \left [\frac {27}{4}\right ], x\right )}{23 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{4}}}+\frac {4 \operatorname {signum}\left (-1+x \right )^{\frac {1}{4}} \left (-\frac {4}{5} x^{2}-\frac {1}{5} x +1\right ) \left (1-x \right )^{\frac {1}{4}}}{9 \left (-\operatorname {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {9}{4}}}\) | \(64\) |
pseudoelliptic | \(\frac {x^{15} \left (\frac {4389 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x}{x}\right ) x^{3}}{32768}-\frac {4389 \ln \left (\frac {x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right ) x^{3}}{32768}-\frac {4389 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right ) x^{3}}{16384}+\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x^{8}-\frac {1}{20} x^{7}-\frac {19}{320} x^{6}-\frac {19}{256} x^{5}-\frac {209}{2048} x^{4}-\frac {1463}{8192} x^{3}-\frac {32}{15} x^{2}-\frac {8}{15} x +\frac {8}{3}\right )\right )}{6 {\left (x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}\right )}^{6} {\left (-\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}+x \right )}^{6} \left (x^{2}+\sqrt {x^{3} \left (-1+x \right )}\right )^{6}}\) | \(161\) |
trager | \(\frac {\left (x^{4}-x^{3}\right )^{\frac {1}{4}} \left (122880 x^{8}-6144 x^{7}-7296 x^{6}-9120 x^{5}-12540 x^{4}-21945 x^{3}-262144 x^{2}-65536 x +327680\right )}{737280 x^{3}}+\frac {1463 \ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-2 x^{3}+x^{2}}{x^{2}}\right )}{65536}-\frac {1463 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}}{x^{2}}\right )}{65536}\) | \(201\) |
risch | \(\frac {\left (122880 x^{9}-129024 x^{8}-1152 x^{7}-1824 x^{6}-3420 x^{5}-9405 x^{4}-240199 x^{3}+196608 x^{2}+393216 x -327680\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{737280 x^{3} \left (-1+x \right )}+\frac {\left (\frac {1463 \ln \left (\frac {2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, x +2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-2 x^{3}+2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x +5 x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}-4 x +1}{\left (-1+x \right )^{2}}\right )}{65536}+\frac {1463 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right )^{2}}\right )}{65536}\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x \left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (-1+x \right )}\) | \(445\) |
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Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=-\frac {131670 \, x^{3} \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 65835 \, x^{3} \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 65835 \, x^{3} \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (122880 \, x^{8} - 6144 \, x^{7} - 7296 \, x^{6} - 9120 \, x^{5} - 12540 \, x^{4} - 21945 \, x^{3} - 262144 \, x^{2} - 65536 \, x + 327680\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{2949120 \, x^{3}} \]
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\[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}{x^{4}}\, dx \]
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\[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=\int { \frac {{\left (x^{8} - 1\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x^{4}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.69 \[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=-\frac {1}{245760} \, {\left (7315 \, {\left (\frac {1}{x} - 1\right )}^{5} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 40755 \, {\left (\frac {1}{x} - 1\right )}^{4} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 92910 \, {\left (\frac {1}{x} - 1\right )}^{3} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 109782 \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 69327 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - 21945 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{6} + \frac {4}{9} \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {4}{5} \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - \frac {1463}{32768} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1463}{65536} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1463}{65536} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {\sqrt [4]{-x^3+x^4} \left (-1+x^8\right )}{x^4} \, dx=\int \frac {\left (x^8-1\right )\,{\left (x^4-x^3\right )}^{1/4}}{x^4} \,d x \]
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