Integrand size = 22, antiderivative size = 20 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^3+x^5}}{x^4} \, dx=\frac {4 \left (-x^3+x^5\right )^{9/4}}{9 x^9} \]
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Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(20)=40\).
Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.95, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2077, 2029, 2057, 372, 371, 2045, 2050} \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^3+x^5}}{x^4} \, dx=\frac {4}{9} \sqrt [4]{x^5-x^3} x-\frac {8 \sqrt [4]{x^5-x^3}}{9 x}+\frac {4 \sqrt [4]{x^5-x^3}}{9 x^3} \]
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Rule 371
Rule 372
Rule 2029
Rule 2045
Rule 2050
Rule 2057
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (\sqrt [4]{-x^3+x^5}-\frac {\sqrt [4]{-x^3+x^5}}{x^4}\right ) \, dx \\ & = \int \sqrt [4]{-x^3+x^5} \, dx-\int \frac {\sqrt [4]{-x^3+x^5}}{x^4} \, dx \\ & = \frac {4 \sqrt [4]{-x^3+x^5}}{9 x^3}+\frac {4}{9} x \sqrt [4]{-x^3+x^5}-\frac {2}{9} \int \frac {x}{\left (-x^3+x^5\right )^{3/4}} \, dx-\frac {2}{9} \int \frac {x^3}{\left (-x^3+x^5\right )^{3/4}} \, dx \\ & = \frac {4 \sqrt [4]{-x^3+x^5}}{9 x^3}-\frac {8 \sqrt [4]{-x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{-x^3+x^5}+\frac {2}{9} \int \frac {x^3}{\left (-x^3+x^5\right )^{3/4}} \, dx-\frac {\left (2 x^{9/4} \left (-1+x^2\right )^{3/4}\right ) \int \frac {x^{3/4}}{\left (-1+x^2\right )^{3/4}} \, dx}{9 \left (-x^3+x^5\right )^{3/4}} \\ & = \frac {4 \sqrt [4]{-x^3+x^5}}{9 x^3}-\frac {8 \sqrt [4]{-x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{-x^3+x^5}-\frac {\left (2 x^{9/4} \left (1-x^2\right )^{3/4}\right ) \int \frac {x^{3/4}}{\left (1-x^2\right )^{3/4}} \, dx}{9 \left (-x^3+x^5\right )^{3/4}}+\frac {\left (2 x^{9/4} \left (-1+x^2\right )^{3/4}\right ) \int \frac {x^{3/4}}{\left (-1+x^2\right )^{3/4}} \, dx}{9 \left (-x^3+x^5\right )^{3/4}} \\ & = \frac {4 \sqrt [4]{-x^3+x^5}}{9 x^3}-\frac {8 \sqrt [4]{-x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{-x^3+x^5}-\frac {8 x^4 \left (1-x^2\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{8},\frac {15}{8},x^2\right )}{63 \left (-x^3+x^5\right )^{3/4}}+\frac {\left (2 x^{9/4} \left (1-x^2\right )^{3/4}\right ) \int \frac {x^{3/4}}{\left (1-x^2\right )^{3/4}} \, dx}{9 \left (-x^3+x^5\right )^{3/4}} \\ & = \frac {4 \sqrt [4]{-x^3+x^5}}{9 x^3}-\frac {8 \sqrt [4]{-x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{-x^3+x^5} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^3+x^5}}{x^4} \, dx=\frac {4 \left (-1+x^2\right )^3}{9 \left (x^3 \left (-1+x^2\right )\right )^{3/4}} \]
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Time = 0.84 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20
method | result | size |
pseudoelliptic | \(\frac {4 \left (x^{2}-1\right )^{2} \left (x^{5}-x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(24\) |
trager | \(\frac {4 \left (x^{4}-2 x^{2}+1\right ) \left (x^{5}-x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(27\) |
gosper | \(\frac {4 \left (x^{5}-x^{3}\right )^{\frac {1}{4}} \left (x^{2}-1\right ) \left (x -1\right ) \left (1+x \right )}{9 x^{3}}\) | \(28\) |
risch | \(\frac {4 \left (x^{3} \left (x^{2}-1\right )\right )^{\frac {1}{4}} \left (x^{6}-3 x^{4}+3 x^{2}-1\right )}{9 x^{3} \left (x^{2}-1\right )}\) | \(39\) |
meijerg | \(\frac {4 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {9}{8}, -\frac {1}{4}\right ], \left [-\frac {1}{8}\right ], x^{2}\right )}{9 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{4}} x^{\frac {9}{4}}}+\frac {4 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{4}} x^{\frac {7}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {7}{8}\right ], \left [\frac {15}{8}\right ], x^{2}\right )}{7 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{4}}}\) | \(66\) |
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^3+x^5}}{x^4} \, dx=\frac {4 \, {\left (x^{5} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 2 \, x^{2} + 1\right )}}{9 \, x^{3}} \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^3+x^5}}{x^4} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{4}}\, dx \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^3+x^5}}{x^4} \, dx=\int { \frac {{\left (x^{5} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{4}} \,d x } \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^3+x^5}}{x^4} \, dx=\int { \frac {{\left (x^{5} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{4}} \,d x } \]
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Time = 5.21 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.35 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^3+x^5}}{x^4} \, dx=\frac {4\,{\left (x^5-x^3\right )}^{1/4}}{9\,x^3}-\frac {8\,{\left (x^5-x^3\right )}^{1/4}}{9\,x}+\frac {4\,x\,{\left (x^5-x^3\right )}^{1/4}}{9} \]
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