Integrand size = 25, antiderivative size = 20 \[ \int x \left (1+2 x^2\right ) \sqrt {-1+2 x^2+2 x^4} \, dx=\frac {1}{6} \left (-1+2 x^2+2 x^4\right )^{3/2} \]
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Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1261, 643} \[ \int x \left (1+2 x^2\right ) \sqrt {-1+2 x^2+2 x^4} \, dx=\frac {1}{6} \left (2 x^4+2 x^2-1\right )^{3/2} \]
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Rule 643
Rule 1261
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int (1+2 x) \sqrt {-1+2 x+2 x^2} \, dx,x,x^2\right ) \\ & = \frac {1}{6} \left (-1+2 x^2+2 x^4\right )^{3/2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int x \left (1+2 x^2\right ) \sqrt {-1+2 x^2+2 x^4} \, dx=\frac {1}{6} \left (-1+2 x^2+2 x^4\right )^{3/2} \]
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Time = 0.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(\frac {\left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{2}}}{6}\) | \(17\) |
default | \(\frac {\left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{2}}}{6}\) | \(17\) |
risch | \(\frac {\left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{2}}}{6}\) | \(17\) |
pseudoelliptic | \(\frac {\left (2 x^{4}+2 x^{2}-1\right )^{\frac {3}{2}}}{6}\) | \(17\) |
trager | \(\left (\frac {1}{3} x^{4}+\frac {1}{3} x^{2}-\frac {1}{6}\right ) \sqrt {2 x^{4}+2 x^{2}-1}\) | \(28\) |
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Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int x \left (1+2 x^2\right ) \sqrt {-1+2 x^2+2 x^4} \, dx=\frac {1}{6} \, {\left (2 \, x^{4} + 2 \, x^{2} - 1\right )}^{\frac {3}{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (15) = 30\).
Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.80 \[ \int x \left (1+2 x^2\right ) \sqrt {-1+2 x^2+2 x^4} \, dx=\frac {x^{4} \sqrt {2 x^{4} + 2 x^{2} - 1}}{3} + \frac {x^{2} \sqrt {2 x^{4} + 2 x^{2} - 1}}{3} - \frac {\sqrt {2 x^{4} + 2 x^{2} - 1}}{6} \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int x \left (1+2 x^2\right ) \sqrt {-1+2 x^2+2 x^4} \, dx=\frac {1}{6} \, {\left (2 \, x^{4} + 2 \, x^{2} - 1\right )}^{\frac {3}{2}} \]
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Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int x \left (1+2 x^2\right ) \sqrt {-1+2 x^2+2 x^4} \, dx=\frac {1}{6} \, {\left (2 \, x^{4} + 2 \, x^{2} - 1\right )}^{\frac {3}{2}} \]
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Time = 5.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int x \left (1+2 x^2\right ) \sqrt {-1+2 x^2+2 x^4} \, dx=\frac {{\left (2\,x^4+2\,x^2-1\right )}^{3/2}}{6} \]
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