Integrand size = 22, antiderivative size = 20 \[ \int \frac {-1+x^4}{x^2 \sqrt [4]{-x^3+x^5}} \, dx=\frac {4 \left (-x^3+x^5\right )^{7/4}}{7 x^7} \]
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Leaf count is larger than twice the leaf count of optimal. \(41\) vs. \(2(20)=40\).
Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.05, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2077, 2050, 2036, 372, 371, 2049} \[ \int \frac {-1+x^4}{x^2 \sqrt [4]{-x^3+x^5}} \, dx=\frac {4 \left (x^5-x^3\right )^{3/4}}{7 x^2}-\frac {4 \left (x^5-x^3\right )^{3/4}}{7 x^4} \]
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Rule 371
Rule 372
Rule 2036
Rule 2049
Rule 2050
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{x^2 \sqrt [4]{-x^3+x^5}}+\frac {x^2}{\sqrt [4]{-x^3+x^5}}\right ) \, dx \\ & = -\int \frac {1}{x^2 \sqrt [4]{-x^3+x^5}} \, dx+\int \frac {x^2}{\sqrt [4]{-x^3+x^5}} \, dx \\ & = -\frac {4 \left (-x^3+x^5\right )^{3/4}}{7 x^4}+\frac {4 \left (-x^3+x^5\right )^{3/4}}{7 x^2} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^4}{x^2 \sqrt [4]{-x^3+x^5}} \, dx=\frac {4 \left (x^3 \left (-1+x^2\right )\right )^{7/4}}{7 x^7} \]
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Time = 0.86 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10
method | result | size |
trager | \(\frac {4 \left (x^{2}-1\right ) \left (x^{5}-x^{3}\right )^{\frac {3}{4}}}{7 x^{4}}\) | \(22\) |
pseudoelliptic | \(\frac {4 \left (x^{2}-1\right ) \left (x^{5}-x^{3}\right )^{\frac {3}{4}}}{7 x^{4}}\) | \(22\) |
risch | \(\frac {\frac {4}{7} x^{4}-\frac {8}{7} x^{2}+\frac {4}{7}}{x \left (x^{3} \left (x^{2}-1\right )\right )^{\frac {1}{4}}}\) | \(27\) |
gosper | \(\frac {4 \left (x^{2}-1\right ) \left (x -1\right ) \left (1+x \right )}{7 \left (x^{5}-x^{3}\right )^{\frac {1}{4}} x}\) | \(28\) |
meijerg | \(\frac {4 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {7}{8}, \frac {1}{4}\right ], \left [\frac {1}{8}\right ], x^{2}\right )}{7 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{4}} x^{\frac {7}{4}}}+\frac {4 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{4}} x^{\frac {9}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {9}{8}\right ], \left [\frac {17}{8}\right ], x^{2}\right )}{9 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{4}}}\) | \(66\) |
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none
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {-1+x^4}{x^2 \sqrt [4]{-x^3+x^5}} \, dx=\frac {4 \, {\left (x^{5} - x^{3}\right )}^{\frac {3}{4}} {\left (x^{2} - 1\right )}}{7 \, x^{4}} \]
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\[ \int \frac {-1+x^4}{x^2 \sqrt [4]{-x^3+x^5}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{2} \sqrt [4]{x^{3} \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
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\[ \int \frac {-1+x^4}{x^2 \sqrt [4]{-x^3+x^5}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{5} - x^{3}\right )}^{\frac {1}{4}} x^{2}} \,d x } \]
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\[ \int \frac {-1+x^4}{x^2 \sqrt [4]{-x^3+x^5}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{5} - x^{3}\right )}^{\frac {1}{4}} x^{2}} \,d x } \]
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Time = 5.44 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.75 \[ \int \frac {-1+x^4}{x^2 \sqrt [4]{-x^3+x^5}} \, dx=\frac {4\,x^2\,{\left (x^5-x^3\right )}^{3/4}-4\,{\left (x^5-x^3\right )}^{3/4}}{7\,x^4} \]
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