Integrand size = 18, antiderivative size = 16 \[ \int \left (2 x+x^3\right ) \left (1+4 x^2+x^4\right ) \, dx=\frac {1}{8} \left (1+4 x^2+x^4\right )^2 \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1602} \[ \int \left (2 x+x^3\right ) \left (1+4 x^2+x^4\right ) \, dx=\frac {1}{8} \left (x^4+4 x^2+1\right )^2 \]
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Rule 1602
Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} \left (1+4 x^2+x^4\right )^2 \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int \left (2 x+x^3\right ) \left (1+4 x^2+x^4\right ) \, dx=x^2+\frac {9 x^4}{4}+x^6+\frac {x^8}{8} \]
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Time = 0.11 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\left (x^{4}+4 x^{2}+1\right )^{2}}{8}\) | \(15\) |
norman | \(\frac {1}{8} x^{8}+x^{6}+\frac {9}{4} x^{4}+x^{2}\) | \(18\) |
parallelrisch | \(\frac {1}{8} x^{8}+x^{6}+\frac {9}{4} x^{4}+x^{2}\) | \(18\) |
risch | \(\frac {1}{8} x^{8}+x^{6}+\frac {9}{4} x^{4}+x^{2}+\frac {1}{8}\) | \(19\) |
gosper | \(\frac {x^{2} \left (x^{6}+8 x^{4}+18 x^{2}+8\right )}{8}\) | \(21\) |
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none
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \left (2 x+x^3\right ) \left (1+4 x^2+x^4\right ) \, dx=\frac {1}{8} \, x^{8} + x^{6} + \frac {9}{4} \, x^{4} + x^{2} \]
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Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \left (2 x+x^3\right ) \left (1+4 x^2+x^4\right ) \, dx=\frac {x^{8}}{8} + x^{6} + \frac {9 x^{4}}{4} + x^{2} \]
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none
Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \left (2 x+x^3\right ) \left (1+4 x^2+x^4\right ) \, dx=\frac {1}{8} \, {\left (x^{4} + 4 \, x^{2} + 1\right )}^{2} \]
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none
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \left (2 x+x^3\right ) \left (1+4 x^2+x^4\right ) \, dx=\frac {1}{4} \, x^{4} + \frac {1}{8} \, {\left (x^{4} + 4 \, x^{2}\right )}^{2} + x^{2} \]
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Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \left (2 x+x^3\right ) \left (1+4 x^2+x^4\right ) \, dx=\frac {x^8}{8}+x^6+\frac {9\,x^4}{4}+x^2 \]
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