Integrand size = 22, antiderivative size = 18 \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=-\frac {4 \left (x+x^5\right )^{3/4}}{1+x^4} \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2081, 460} \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=-\frac {4 x}{\sqrt [4]{x^5+x}} \]
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Rule 460
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {-3+x^4}{\sqrt [4]{x} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}} \\ & = -\frac {4 x}{\sqrt [4]{x+x^5}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 2 in optimal.
Time = 10.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.33 \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4 \sqrt [4]{1+x^4} \left (-19 x \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {5}{4},\frac {19}{16},-x^4\right )+x^5 \operatorname {Hypergeometric2F1}\left (\frac {19}{16},\frac {5}{4},\frac {35}{16},-x^4\right )\right )}{19 \sqrt [4]{x+x^5}} \]
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Time = 1.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(-\frac {4 x}{\left (x^{5}+x \right )^{\frac {1}{4}}}\) | \(11\) |
risch | \(-\frac {4 x}{{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}\) | \(13\) |
pseudoelliptic | \(-\frac {4 x}{{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}\) | \(13\) |
trager | \(-\frac {4 \left (x^{5}+x \right )^{\frac {3}{4}}}{x^{4}+1}\) | \(17\) |
meijerg | \(-4 x^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [\frac {3}{16}, \frac {5}{4}\right ], \left [\frac {19}{16}\right ], -x^{4}\right )+\frac {4 x^{\frac {19}{4}} \operatorname {hypergeom}\left (\left [\frac {19}{16}, \frac {5}{4}\right ], \left [\frac {35}{16}\right ], -x^{4}\right )}{19}\) | \(34\) |
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none
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=-\frac {4 \, {\left (x^{5} + x\right )}^{\frac {3}{4}}}{x^{4} + 1} \]
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\[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int \frac {x^{4} - 3}{\sqrt [4]{x \left (x^{4} + 1\right )} \left (x^{4} + 1\right )}\, dx \]
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\[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int { \frac {x^{4} - 3}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]
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\[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int { \frac {x^{4} - 3}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]
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Time = 4.96 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx=-\frac {4\,{\left (x^5+x\right )}^{3/4}}{x^4+1} \]
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