Integrand size = 20, antiderivative size = 18 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{x^4} \, dx=\frac {2 \left (x^2+x^6\right )^{5/4}}{5 x^5} \]
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Time = 0.01 (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 = {1604} \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{x^4} \, dx=\frac {2 \left (x^6+x^2\right )^{5/4}}{5 x^5} \]
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Rule 1604
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (x^2+x^6\right )^{5/4}}{5 x^5} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{x^4} \, dx=\frac {2 \left (1+x^4\right ) \sqrt [4]{x^2+x^6}}{5 x^3} \]
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Time = 0.86 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11
method | result | size |
gosper | \(\frac {2 \left (x^{4}+1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}\) | \(20\) |
trager | \(\frac {2 \left (x^{4}+1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}\) | \(20\) |
pseudoelliptic | \(\frac {2 \left (x^{4}+1\right ) \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}{5 x^{3}}\) | \(22\) |
meijerg | \(\frac {2 \operatorname {hypergeom}\left (\left [-\frac {5}{8}, -\frac {1}{4}\right ], \left [\frac {3}{8}\right ], -x^{4}\right )}{5 x^{\frac {5}{2}}}+\frac {2 x^{\frac {3}{2}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {3}{8}\right ], \left [\frac {11}{8}\right ], -x^{4}\right )}{3}\) | \(34\) |
risch | \(\frac {2 \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}} \left (x^{8}+2 x^{4}+1\right )}{5 x^{3} \left (x^{4}+1\right )}\) | \(34\) |
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none
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{x^4} \, dx=\frac {2 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{5 \, x^{3}} \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{x^4} \, dx=\int \frac {\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{4}}\, dx \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{x^4} \, dx=\int { \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{4}} \,d x } \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{x^4} \, dx=\int { \frac {{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{4}} \,d x } \]
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Time = 5.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^2+x^6}}{x^4} \, dx=\frac {2\,{\left (x^6+x^2\right )}^{1/4}\,\left (x^4+1\right )}{5\,x^3} \]
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