Integrand size = 24, antiderivative size = 24 \[ \int \frac {1+2 x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {5+x+x^2}}{\sqrt {2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1038, 212} \[ \int \frac {1+2 x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {x^2+x+5}}{\sqrt {2}}\right ) \]
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Rule 212
Rule 1038
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {5+x+x^2}\right )\right ) \\ & = -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {5+x+x^2}}{\sqrt {2}}\right ) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1+2 x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {5+x+x^2}}{\sqrt {2}}\right ) \]
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Time = 1.78 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\operatorname {arctanh}\left (\frac {\sqrt {x^{2}+x +5}\, \sqrt {2}}{2}\right ) \sqrt {2}\) | \(20\) |
pseudoelliptic | \(-\operatorname {arctanh}\left (\frac {\sqrt {x^{2}+x +5}\, \sqrt {2}}{2}\right ) \sqrt {2}\) | \(20\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {x^{2}+x +5}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{2}+x +3}\right )}{2}\) | \(56\) |
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none
Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {1+2 x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {x^{2} - 2 \, \sqrt {2} \sqrt {x^{2} + x + 5} + x + 7}{x^{2} + x + 3}\right ) \]
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Time = 1.74 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {1+2 x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=\frac {\sqrt {2} \left (\log {\left (\sqrt {x^{2} + x + 5} - \sqrt {2} \right )} - \log {\left (\sqrt {x^{2} + x + 5} + \sqrt {2} \right )}\right )}{2} \]
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\[ \int \frac {1+2 x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=\int { \frac {2 \, x + 1}{\sqrt {x^{2} + x + 5} {\left (x^{2} + x + 3\right )}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {1+2 x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (\sqrt {2} + \sqrt {x^{2} + x + 5}\right ) + \frac {1}{2} \, \sqrt {2} \log \left (-\sqrt {2} + \sqrt {x^{2} + x + 5}\right ) \]
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Time = 12.55 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {1+2 x}{\left (3+x+x^2\right ) \sqrt {5+x+x^2}} \, dx=-\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {x^2+x+5}}{2}\right ) \]
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