Integrand size = 40, antiderivative size = 23 \[ \int \frac {x^2 \left (3+2 x^2\right ) \left (1+x^2+2 x^6\right )}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^6}} \, dx=\frac {x^3 \sqrt {1+x^2+x^6}}{1+x^2} \]
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Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {1604} \[ \int \frac {x^2 \left (3+2 x^2\right ) \left (1+x^2+2 x^6\right )}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^6}} \, dx=\frac {x^3 \sqrt {x^6+x^2+1}}{x^2+1} \]
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Rule 1604
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \sqrt {1+x^2+x^6}}{1+x^2} \\ \end{align*}
Time = 3.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (3+2 x^2\right ) \left (1+x^2+2 x^6\right )}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^6}} \, dx=\frac {x^3 \sqrt {1+x^2+x^6}}{1+x^2} \]
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Time = 1.50 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96
method | result | size |
gosper | \(\frac {x^{3} \sqrt {x^{6}+x^{2}+1}}{x^{2}+1}\) | \(22\) |
trager | \(\frac {x^{3} \sqrt {x^{6}+x^{2}+1}}{x^{2}+1}\) | \(22\) |
risch | \(\frac {x^{3} \sqrt {x^{6}+x^{2}+1}}{x^{2}+1}\) | \(22\) |
pseudoelliptic | \(\frac {x^{3} \sqrt {x^{6}+x^{2}+1}}{x^{2}+1}\) | \(22\) |
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none
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \left (3+2 x^2\right ) \left (1+x^2+2 x^6\right )}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^6}} \, dx=\frac {\sqrt {x^{6} + x^{2} + 1} x^{3}}{x^{2} + 1} \]
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\[ \int \frac {x^2 \left (3+2 x^2\right ) \left (1+x^2+2 x^6\right )}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^6}} \, dx=\int \frac {x^{2} \cdot \left (2 x^{2} + 3\right ) \left (2 x^{6} + x^{2} + 1\right )}{\left (x^{2} + 1\right )^{2} \sqrt {x^{6} + x^{2} + 1}}\, dx \]
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\[ \int \frac {x^2 \left (3+2 x^2\right ) \left (1+x^2+2 x^6\right )}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^6}} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{2} + 1\right )} {\left (2 \, x^{2} + 3\right )} x^{2}}{\sqrt {x^{6} + x^{2} + 1} {\left (x^{2} + 1\right )}^{2}} \,d x } \]
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\[ \int \frac {x^2 \left (3+2 x^2\right ) \left (1+x^2+2 x^6\right )}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^6}} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{2} + 1\right )} {\left (2 \, x^{2} + 3\right )} x^{2}}{\sqrt {x^{6} + x^{2} + 1} {\left (x^{2} + 1\right )}^{2}} \,d x } \]
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Time = 5.68 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {x^2 \left (3+2 x^2\right ) \left (1+x^2+2 x^6\right )}{\left (1+x^2\right )^2 \sqrt {1+x^2+x^6}} \, dx=\frac {x^3\,\sqrt {x^6+x^2+1}}{x^2+1} \]
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