Integrand size = 20, antiderivative size = 18 \[ \int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx=\frac {3 \left (x^2+x^4\right )^{2/3}}{2 x^2} \]
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Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2059, 777} \[ \int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx=\frac {3 \left (x^4+x^2\right )^{2/3}}{2 x^2} \]
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Rule 777
Rule 2059
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {-1+x}{x \sqrt [3]{x+x^2}} \, dx,x,x^2\right ) \\ & = \frac {3 \left (x^2+x^4\right )^{2/3}}{2 x^2} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx=\frac {3 \left (1+x^2\right )}{2 \sqrt [3]{x^2+x^4}} \]
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Time = 0.81 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
trager | \(\frac {3 \left (x^{4}+x^{2}\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(15\) |
gosper | \(\frac {\frac {3 x^{2}}{2}+\frac {3}{2}}{\left (x^{4}+x^{2}\right )^{\frac {1}{3}}}\) | \(17\) |
pseudoelliptic | \(\frac {3 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(17\) |
risch | \(\frac {\frac {3 x^{2}}{2}+\frac {3}{2}}{\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}\) | \(19\) |
meijerg | \(\frac {3 \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {2}{3}\right ], -x^{2}\right )}{2 x^{\frac {2}{3}}}+\frac {3 x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{2}\right )}{4}\) | \(34\) |
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none
Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx=\frac {3 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]
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\[ \int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{x \sqrt [3]{x^{2} \left (x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x} \,d x } \]
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\[ \int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x} \,d x } \]
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Time = 5.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx=\frac {3\,{\left (x^4+x^2\right )}^{2/3}}{2\,x^2} \]
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