Integrand size = 27, antiderivative size = 16 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^5}}{x^2 \left (1+x^2\right )} \, dx=\frac {4 \sqrt [4]{x^3+x^5}}{x} \]
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Time = 0.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2081, 460} \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^5}}{x^2 \left (1+x^2\right )} \, dx=\frac {4 \sqrt [4]{x^5+x^3}}{x} \]
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Rule 460
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{x^3+x^5} \int \frac {-1+x^2}{x^{5/4} \left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}} \\ & = \frac {4 \sqrt [4]{x^3+x^5}}{x} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^5}}{x^2 \left (1+x^2\right )} \, dx=\frac {4 \sqrt [4]{x^3+x^5}}{x} \]
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Time = 1.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(\frac {4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{x}\) | \(15\) |
trager | \(\frac {4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{x}\) | \(15\) |
risch | \(\frac {4 \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\) | \(17\) |
pseudoelliptic | \(\frac {4 \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\) | \(17\) |
meijerg | \(\frac {4 \operatorname {hypergeom}\left (\left [-\frac {1}{8}, \frac {3}{4}\right ], \left [\frac {7}{8}\right ], -x^{2}\right )}{x^{\frac {1}{4}}}+\frac {4 x^{\frac {7}{4}} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {7}{8}\right ], \left [\frac {15}{8}\right ], -x^{2}\right )}{7}\) | \(34\) |
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none
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^5}}{x^2 \left (1+x^2\right )} \, dx=\frac {4 \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}}{x} \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^5}}{x^2 \left (1+x^2\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{x^{2} \left (x^{2} + 1\right )}\, dx \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^5}}{x^2 \left (1+x^2\right )} \, dx=\int { \frac {{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}} \,d x } \]
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\[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^5}}{x^2 \left (1+x^2\right )} \, dx=\int { \frac {{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}} \,d x } \]
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Time = 4.98 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^5}}{x^2 \left (1+x^2\right )} \, dx=\frac {4\,{\left (x^5+x^3\right )}^{1/4}}{x} \]
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