Integrand size = 23, antiderivative size = 79 \[ \int \frac {x^2}{1+2 x+2 \sqrt {1+x+x^2}} \, dx=-\frac {x^3}{9}-\frac {x^4}{6}+\frac {1}{96} (1+2 x) \sqrt {1+x+x^2}-\frac {5}{36} \left (1+x+x^2\right )^{3/2}+\frac {1}{6} x \left (1+x+x^2\right )^{3/2}+\frac {1}{64} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6874, 756, 654, 626, 633, 221} \[ \int \frac {x^2}{1+2 x+2 \sqrt {1+x+x^2}} \, dx=\frac {1}{64} \text {arcsinh}\left (\frac {2 x+1}{\sqrt {3}}\right )-\frac {x^4}{6}-\frac {x^3}{9}+\frac {1}{6} \left (x^2+x+1\right )^{3/2} x-\frac {5}{36} \left (x^2+x+1\right )^{3/2}+\frac {1}{96} (2 x+1) \sqrt {x^2+x+1} \]
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Rule 221
Rule 626
Rule 633
Rule 654
Rule 756
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {x^2}{3}-\frac {2 x^3}{3}+\frac {2}{3} x^2 \sqrt {1+x+x^2}\right ) \, dx \\ & = -\frac {x^3}{9}-\frac {x^4}{6}+\frac {2}{3} \int x^2 \sqrt {1+x+x^2} \, dx \\ & = -\frac {x^3}{9}-\frac {x^4}{6}+\frac {1}{6} x \left (1+x+x^2\right )^{3/2}+\frac {1}{6} \int \left (-1-\frac {5 x}{2}\right ) \sqrt {1+x+x^2} \, dx \\ & = -\frac {x^3}{9}-\frac {x^4}{6}-\frac {5}{36} \left (1+x+x^2\right )^{3/2}+\frac {1}{6} x \left (1+x+x^2\right )^{3/2}+\frac {1}{24} \int \sqrt {1+x+x^2} \, dx \\ & = -\frac {x^3}{9}-\frac {x^4}{6}+\frac {1}{96} (1+2 x) \sqrt {1+x+x^2}-\frac {5}{36} \left (1+x+x^2\right )^{3/2}+\frac {1}{6} x \left (1+x+x^2\right )^{3/2}+\frac {1}{64} \int \frac {1}{\sqrt {1+x+x^2}} \, dx \\ & = -\frac {x^3}{9}-\frac {x^4}{6}+\frac {1}{96} (1+2 x) \sqrt {1+x+x^2}-\frac {5}{36} \left (1+x+x^2\right )^{3/2}+\frac {1}{6} x \left (1+x+x^2\right )^{3/2}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right )}{64 \sqrt {3}} \\ & = -\frac {x^3}{9}-\frac {x^4}{6}+\frac {1}{96} (1+2 x) \sqrt {1+x+x^2}-\frac {5}{36} \left (1+x+x^2\right )^{3/2}+\frac {1}{6} x \left (1+x+x^2\right )^{3/2}+\frac {1}{64} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right ) \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.81 \[ \int \frac {x^2}{1+2 x+2 \sqrt {1+x+x^2}} \, dx=-\frac {1}{18} x^3 (2+3 x)+\frac {1}{288} \sqrt {1+x+x^2} \left (-37+14 x+8 x^2+48 x^3\right )-\frac {1}{64} \log \left (-1-2 x+2 \sqrt {1+x+x^2}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70
method | result | size |
trager | \(-\frac {\left (2+3 x \right ) x^{3}}{18}+\frac {\left (\frac {1}{2} x^{3}+\frac {1}{12} x^{2}+\frac {7}{48} x -\frac {37}{96}\right ) \sqrt {x^{2}+x +1}}{3}+\frac {\ln \left (1+2 x +2 \sqrt {x^{2}+x +1}\right )}{64}\) | \(55\) |
default | \(-\frac {x^{3}}{9}-\frac {x^{4}}{6}+\frac {x \left (x^{2}+x +1\right )^{\frac {3}{2}}}{6}-\frac {5 \left (x^{2}+x +1\right )^{\frac {3}{2}}}{36}+\frac {\left (1+2 x \right ) \sqrt {x^{2}+x +1}}{96}+\frac {\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{64}\) | \(59\) |
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Time = 0.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.68 \[ \int \frac {x^2}{1+2 x+2 \sqrt {1+x+x^2}} \, dx=-\frac {1}{6} \, x^{4} - \frac {1}{9} \, x^{3} + \frac {1}{288} \, {\left (48 \, x^{3} + 8 \, x^{2} + 14 \, x - 37\right )} \sqrt {x^{2} + x + 1} - \frac {1}{64} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]
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\[ \int \frac {x^2}{1+2 x+2 \sqrt {1+x+x^2}} \, dx=\int \frac {x^{2}}{2 x + 2 \sqrt {x^{2} + x + 1} + 1}\, dx \]
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\[ \int \frac {x^2}{1+2 x+2 \sqrt {1+x+x^2}} \, dx=\int { \frac {x^{2}}{2 \, x + 2 \, \sqrt {x^{2} + x + 1} + 1} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.68 \[ \int \frac {x^2}{1+2 x+2 \sqrt {1+x+x^2}} \, dx=-\frac {1}{6} \, x^{4} - \frac {1}{9} \, x^{3} + \frac {1}{288} \, {\left (2 \, {\left (4 \, {\left (6 \, x + 1\right )} x + 7\right )} x - 37\right )} \sqrt {x^{2} + x + 1} - \frac {1}{64} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]
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Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \frac {x^2}{1+2 x+2 \sqrt {1+x+x^2}} \, dx=\frac {\ln \left (x+\sqrt {x^2+x+1}+\frac {1}{2}\right )}{64}-\frac {\left (\frac {x}{2}+\frac {1}{4}\right )\,\sqrt {x^2+x+1}}{6}-\frac {x^3}{9}-\frac {x^4}{6}-\frac {5\,\left (8\,x^2+2\,x+5\right )\,\sqrt {x^2+x+1}}{288}+\frac {x\,{\left (x^2+x+1\right )}^{3/2}}{6} \]
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