Integrand size = 20, antiderivative size = 18 \[ \int \frac {-1+x^2}{x \sqrt [4]{x^3+x^5}} \, dx=\frac {4 \left (x^3+x^5\right )^{3/4}}{3 x^3} \]
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Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2061} \[ \int \frac {-1+x^2}{x \sqrt [4]{x^3+x^5}} \, dx=\frac {4 \left (x^5+x^3\right )^{3/4}}{3 x^3} \]
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Rule 2061
Rubi steps \begin{align*} \text {integral}& = \frac {4 \left (x^3+x^5\right )^{3/4}}{3 x^3} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-1+x^2}{x \sqrt [4]{x^3+x^5}} \, dx=\frac {4 \left (1+x^2\right )}{3 \sqrt [4]{x^3+x^5}} \]
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Time = 0.82 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
trager | \(\frac {4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(15\) |
gosper | \(\frac {\frac {4 x^{2}}{3}+\frac {4}{3}}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\) | \(17\) |
pseudoelliptic | \(\frac {4 \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(17\) |
risch | \(\frac {\frac {4 x^{2}}{3}+\frac {4}{3}}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\) | \(19\) |
meijerg | \(\frac {4 \operatorname {hypergeom}\left (\left [-\frac {3}{8}, \frac {1}{4}\right ], \left [\frac {5}{8}\right ], -x^{2}\right )}{3 x^{\frac {3}{4}}}+\frac {4 x^{\frac {5}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {5}{8}\right ], \left [\frac {13}{8}\right ], -x^{2}\right )}{5}\) | \(34\) |
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none
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-1+x^2}{x \sqrt [4]{x^3+x^5}} \, dx=\frac {4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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\[ \int \frac {-1+x^2}{x \sqrt [4]{x^3+x^5}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{x \sqrt [4]{x^{3} \left (x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {-1+x^2}{x \sqrt [4]{x^3+x^5}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x} \,d x } \]
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\[ \int \frac {-1+x^2}{x \sqrt [4]{x^3+x^5}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x} \,d x } \]
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Time = 5.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-1+x^2}{x \sqrt [4]{x^3+x^5}} \, dx=\frac {4\,{\left (x^5+x^3\right )}^{3/4}}{3\,x^3} \]
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