Integrand size = 13, antiderivative size = 16 \[ \int \frac {\sqrt [3]{x+x^3}}{x^4} \, dx=-\frac {3 \left (x+x^3\right )^{4/3}}{8 x^4} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2039} \[ \int \frac {\sqrt [3]{x+x^3}}{x^4} \, dx=-\frac {3 \left (x^3+x\right )^{4/3}}{8 x^4} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \left (x+x^3\right )^{4/3}}{8 x^4} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt [3]{x+x^3}}{x^4} \, dx=-\frac {3 \left (1+x^2\right ) \sqrt [3]{x+x^3}}{8 x^3} \]
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Time = 0.80 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
meijerg | \(-\frac {3 \left (x^{2}+1\right )^{\frac {4}{3}}}{8 x^{\frac {8}{3}}}\) | \(13\) |
gosper | \(-\frac {3 \left (x^{2}+1\right ) \left (x^{3}+x \right )^{\frac {1}{3}}}{8 x^{3}}\) | \(18\) |
trager | \(-\frac {3 \left (x^{2}+1\right ) \left (x^{3}+x \right )^{\frac {1}{3}}}{8 x^{3}}\) | \(18\) |
pseudoelliptic | \(-\frac {3 \left (x^{2}+1\right ) {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}}{8 x^{3}}\) | \(20\) |
risch | \(-\frac {3 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}} \left (x^{4}+2 x^{2}+1\right )}{8 x^{3} \left (x^{2}+1\right )}\) | \(32\) |
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none
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt [3]{x+x^3}}{x^4} \, dx=-\frac {3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{8 \, x^{3}} \]
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\[ \int \frac {\sqrt [3]{x+x^3}}{x^4} \, dx=\int \frac {\sqrt [3]{x \left (x^{2} + 1\right )}}{x^{4}}\, dx \]
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none
Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt [3]{x+x^3}}{x^4} \, dx=-\frac {3 \, {\left (x^{3} + x\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}{8 \, x^{\frac {11}{3}}} \]
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none
Time = 0.30 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt [3]{x+x^3}}{x^4} \, dx=-\frac {3}{8} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} \]
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Time = 5.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {\sqrt [3]{x+x^3}}{x^4} \, dx=-\frac {3\,{\left (x^3+x\right )}^{1/3}+3\,x^2\,{\left (x^3+x\right )}^{1/3}}{8\,x^3} \]
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