Integrand size = 30, antiderivative size = 30 \[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \sqrt [4]{-x+x^5}} \, dx=\frac {4 \left (-3-7 x^3+3 x^4\right ) \left (-x+x^5\right )^{3/4}}{21 x^6} \]
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Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.83, number of steps used = 21, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2077, 2050, 2057, 372, 371} \[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \sqrt [4]{-x+x^5}} \, dx=-\frac {4 \left (x^5-x\right )^{3/4}}{7 x^6}-\frac {4 \left (x^5-x\right )^{3/4}}{3 x^3}+\frac {4 \left (x^5-x\right )^{3/4}}{7 x^2} \]
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Rule 371
Rule 372
Rule 2050
Rule 2057
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3}{x^6 \sqrt [4]{-x+x^5}}-\frac {3}{x^3 \sqrt [4]{-x+x^5}}+\frac {2}{x^2 \sqrt [4]{-x+x^5}}-\frac {x}{\sqrt [4]{-x+x^5}}+\frac {x^2}{\sqrt [4]{-x+x^5}}\right ) \, dx \\ & = 2 \int \frac {1}{x^2 \sqrt [4]{-x+x^5}} \, dx-3 \int \frac {1}{x^6 \sqrt [4]{-x+x^5}} \, dx-3 \int \frac {1}{x^3 \sqrt [4]{-x+x^5}} \, dx-\int \frac {x}{\sqrt [4]{-x+x^5}} \, dx+\int \frac {x^2}{\sqrt [4]{-x+x^5}} \, dx \\ & = -\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^6}-\frac {4 \left (-x+x^5\right )^{3/4}}{3 x^3}+\frac {8 \left (-x+x^5\right )^{3/4}}{5 x^2}-\frac {9}{7} \int \frac {1}{x^2 \sqrt [4]{-x+x^5}} \, dx-\frac {14}{5} \int \frac {x^2}{\sqrt [4]{-x+x^5}} \, dx-\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \int \frac {x^{3/4}}{\sqrt [4]{-1+x^4}} \, dx}{\sqrt [4]{-x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{-1+x^4}} \, dx}{\sqrt [4]{-x+x^5}}+\int \frac {x}{\sqrt [4]{-x+x^5}} \, dx \\ & = -\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^6}-\frac {4 \left (-x+x^5\right )^{3/4}}{3 x^3}+\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^2}+\frac {9}{5} \int \frac {x^2}{\sqrt [4]{-x+x^5}} \, dx-\frac {\left (\sqrt [4]{x} \sqrt [4]{1-x^4}\right ) \int \frac {x^{3/4}}{\sqrt [4]{1-x^4}} \, dx}{\sqrt [4]{-x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{1-x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1-x^4}} \, dx}{\sqrt [4]{-x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \int \frac {x^{3/4}}{\sqrt [4]{-1+x^4}} \, dx}{\sqrt [4]{-x+x^5}}-\frac {\left (14 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{-1+x^4}} \, dx}{5 \sqrt [4]{-x+x^5}} \\ & = -\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^6}-\frac {4 \left (-x+x^5\right )^{3/4}}{3 x^3}+\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^2}-\frac {4 x^2 \sqrt [4]{1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {7}{16},\frac {23}{16},x^4\right )}{7 \sqrt [4]{-x+x^5}}+\frac {4 x^3 \sqrt [4]{1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{16},\frac {27}{16},x^4\right )}{11 \sqrt [4]{-x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{1-x^4}\right ) \int \frac {x^{3/4}}{\sqrt [4]{1-x^4}} \, dx}{\sqrt [4]{-x+x^5}}-\frac {\left (14 \sqrt [4]{x} \sqrt [4]{1-x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1-x^4}} \, dx}{5 \sqrt [4]{-x+x^5}}+\frac {\left (9 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{-1+x^4}} \, dx}{5 \sqrt [4]{-x+x^5}} \\ & = -\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^6}-\frac {4 \left (-x+x^5\right )^{3/4}}{3 x^3}+\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^2}-\frac {36 x^3 \sqrt [4]{1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{16},\frac {27}{16},x^4\right )}{55 \sqrt [4]{-x+x^5}}+\frac {\left (9 \sqrt [4]{x} \sqrt [4]{1-x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1-x^4}} \, dx}{5 \sqrt [4]{-x+x^5}} \\ & = -\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^6}-\frac {4 \left (-x+x^5\right )^{3/4}}{3 x^3}+\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 2 in optimal.
Time = 10.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.90 \[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \sqrt [4]{-x+x^5}} \, dx=\frac {4 \sqrt [4]{1-x^4} \left (165 \operatorname {Hypergeometric2F1}\left (-\frac {21}{16},\frac {1}{4},-\frac {5}{16},x^4\right )+x^3 \left (385 \operatorname {Hypergeometric2F1}\left (-\frac {9}{16},\frac {1}{4},\frac {7}{16},x^4\right )-462 x \operatorname {Hypergeometric2F1}\left (-\frac {5}{16},\frac {1}{4},\frac {11}{16},x^4\right )-165 x^4 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {7}{16},\frac {23}{16},x^4\right )+105 x^5 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{16},\frac {27}{16},x^4\right )\right )\right )}{1155 x^5 \sqrt [4]{x \left (-1+x^4\right )}} \]
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Time = 0.90 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90
method | result | size |
trager | \(\frac {4 \left (3 x^{4}-7 x^{3}-3\right ) \left (x^{5}-x \right )^{\frac {3}{4}}}{21 x^{6}}\) | \(27\) |
pseudoelliptic | \(\frac {4 \left (3 x^{4}-7 x^{3}-3\right ) \left (x^{5}-x \right )^{\frac {3}{4}}}{21 x^{6}}\) | \(27\) |
risch | \(\frac {\frac {4}{7} x^{8}-\frac {8}{7} x^{4}+\frac {4}{7}-\frac {4}{3} x^{7}+\frac {4}{3} x^{3}}{x^{5} {\left (x \left (x^{4}-1\right )\right )}^{\frac {1}{4}}}\) | \(37\) |
gosper | \(\frac {4 \left (x^{2}+1\right ) \left (x -1\right ) \left (1+x \right ) \left (3 x^{4}-7 x^{3}-3\right )}{21 x^{5} \left (x^{5}-x \right )^{\frac {1}{4}}}\) | \(38\) |
meijerg | \(\frac {4 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {21}{16}, \frac {1}{4}\right ], \left [-\frac {5}{16}\right ], x^{4}\right )}{7 \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{\frac {21}{4}}}-\frac {8 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {5}{16}, \frac {1}{4}\right ], \left [\frac {11}{16}\right ], x^{4}\right )}{5 \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{\frac {5}{4}}}+\frac {4 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {9}{16}, \frac {1}{4}\right ], \left [\frac {7}{16}\right ], x^{4}\right )}{3 \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{\frac {9}{4}}}+\frac {4 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} x^{\frac {11}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {11}{16}\right ], \left [\frac {27}{16}\right ], x^{4}\right )}{11 \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}-\frac {4 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{4}} x^{\frac {7}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {7}{16}\right ], \left [\frac {23}{16}\right ], x^{4}\right )}{7 \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\) | \(162\) |
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Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \sqrt [4]{-x+x^5}} \, dx=\frac {4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}} {\left (3 \, x^{4} - 7 \, x^{3} - 3\right )}}{21 \, x^{6}} \]
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\[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \sqrt [4]{-x+x^5}} \, dx=\int \frac {\left (x^{4} + 3\right ) \left (x^{4} - x^{3} - 1\right )}{x^{6} \sqrt [4]{x \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \sqrt [4]{-x+x^5}} \, dx=\int { \frac {{\left (x^{4} - x^{3} - 1\right )} {\left (x^{4} + 3\right )}}{{\left (x^{5} - x\right )}^{\frac {1}{4}} x^{6}} \,d x } \]
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\[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \sqrt [4]{-x+x^5}} \, dx=\int { \frac {{\left (x^{4} - x^{3} - 1\right )} {\left (x^{4} + 3\right )}}{{\left (x^{5} - x\right )}^{\frac {1}{4}} x^{6}} \,d x } \]
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Time = 5.32 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \sqrt [4]{-x+x^5}} \, dx=-\frac {12\,{\left (x^5-x\right )}^{3/4}+28\,x^3\,{\left (x^5-x\right )}^{3/4}-12\,x^4\,{\left (x^5-x\right )}^{3/4}}{21\,x^6} \]
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