Integrand size = 22, antiderivative size = 16 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^4}}{\left (1+x^3\right )^2} \, dx=-\frac {3 x^2}{2 \left (x+x^4\right )^{2/3}} \]
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Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2081, 460} \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^4}}{\left (1+x^3\right )^2} \, dx=-\frac {3 x \sqrt [3]{x^4+x}}{2 \left (x^3+1\right )} \]
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Rule 460
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{x+x^4} \int \frac {\sqrt [3]{x} \left (-2+x^3\right )}{\left (1+x^3\right )^{5/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^3}} \\ & = -\frac {3 x \sqrt [3]{x+x^4}}{2 \left (1+x^3\right )} \\ \end{align*}
Time = 6.63 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^4}}{\left (1+x^3\right )^2} \, dx=-\frac {3 x \sqrt [3]{x+x^4}}{2 \left (1+x^3\right )} \]
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Time = 1.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(-\frac {3 x^{2}}{2 \left (x^{4}+x \right )^{\frac {2}{3}}}\) | \(13\) |
gosper | \(-\frac {3 x \left (x^{4}+x \right )^{\frac {1}{3}}}{2 \left (x^{3}+1\right )}\) | \(18\) |
trager | \(-\frac {3 x \left (x^{4}+x \right )^{\frac {1}{3}}}{2 \left (x^{3}+1\right )}\) | \(18\) |
risch | \(-\frac {3 {\left (x \left (x^{3}+1\right )\right )}^{\frac {1}{3}} x}{2 \left (x^{3}+1\right )}\) | \(20\) |
meijerg | \(-\frac {3 x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {4}{9}, \frac {5}{3}\right ], \left [\frac {13}{9}\right ], -x^{3}\right )}{2}+\frac {3 x^{\frac {13}{3}} \operatorname {hypergeom}\left (\left [\frac {13}{9}, \frac {5}{3}\right ], \left [\frac {22}{9}\right ], -x^{3}\right )}{13}\) | \(34\) |
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^4}}{\left (1+x^3\right )^2} \, dx=-\frac {3 \, {\left (x^{4} + x\right )}^{\frac {1}{3}} x}{2 \, {\left (x^{3} + 1\right )}} \]
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\[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^4}}{\left (1+x^3\right )^2} \, dx=\int \frac {\sqrt [3]{x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x^{3} - 2\right )}{\left (x + 1\right )^{2} \left (x^{2} - x + 1\right )^{2}}\, dx \]
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\[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^4}}{\left (1+x^3\right )^2} \, dx=\int { \frac {{\left (x^{4} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{{\left (x^{3} + 1\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^4}}{\left (1+x^3\right )^2} \, dx=\int { \frac {{\left (x^{4} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{{\left (x^{3} + 1\right )}^{2}} \,d x } \]
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Time = 4.97 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^4}}{\left (1+x^3\right )^2} \, dx=-\frac {3\,x\,{\left (x^4+x\right )}^{1/3}}{2\,\left (x^3+1\right )} \]
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