Integrand size = 14, antiderivative size = 53 \[ \int \frac {x^3}{\sqrt {1+x+x^2}} \, dx=\frac {1}{3} x^2 \sqrt {1+x+x^2}-\frac {1}{24} (1+10 x) \sqrt {1+x+x^2}+\frac {7}{16} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {756, 793, 633, 221} \[ \int \frac {x^3}{\sqrt {1+x+x^2}} \, dx=\frac {7}{16} \text {arcsinh}\left (\frac {2 x+1}{\sqrt {3}}\right )+\frac {1}{3} \sqrt {x^2+x+1} x^2-\frac {1}{24} (10 x+1) \sqrt {x^2+x+1} \]
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Rule 221
Rule 633
Rule 756
Rule 793
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^2 \sqrt {1+x+x^2}+\frac {1}{3} \int \frac {\left (-2-\frac {5 x}{2}\right ) x}{\sqrt {1+x+x^2}} \, dx \\ & = \frac {1}{3} x^2 \sqrt {1+x+x^2}-\frac {1}{24} (1+10 x) \sqrt {1+x+x^2}+\frac {7}{16} \int \frac {1}{\sqrt {1+x+x^2}} \, dx \\ & = \frac {1}{3} x^2 \sqrt {1+x+x^2}-\frac {1}{24} (1+10 x) \sqrt {1+x+x^2}+\frac {7 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right )}{16 \sqrt {3}} \\ & = \frac {1}{3} x^2 \sqrt {1+x+x^2}-\frac {1}{24} (1+10 x) \sqrt {1+x+x^2}+\frac {7}{16} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.89 \[ \int \frac {x^3}{\sqrt {1+x+x^2}} \, dx=\frac {1}{24} \sqrt {1+x+x^2} \left (-1-10 x+8 x^2\right )-\frac {7}{16} \log \left (-1-2 x+2 \sqrt {1+x+x^2}\right ) \]
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Time = 0.42 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.62
method | result | size |
risch | \(\frac {\left (8 x^{2}-10 x -1\right ) \sqrt {x^{2}+x +1}}{24}+\frac {7 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{16}\) | \(33\) |
trager | \(\left (\frac {1}{3} x^{2}-\frac {5}{12} x -\frac {1}{24}\right ) \sqrt {x^{2}+x +1}-\frac {7 \ln \left (2 \sqrt {x^{2}+x +1}-1-2 x \right )}{16}\) | \(39\) |
default | \(\frac {x^{2} \sqrt {x^{2}+x +1}}{3}-\frac {5 x \sqrt {x^{2}+x +1}}{12}-\frac {\sqrt {x^{2}+x +1}}{24}+\frac {7 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{16}\) | \(47\) |
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Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.74 \[ \int \frac {x^3}{\sqrt {1+x+x^2}} \, dx=\frac {1}{24} \, {\left (8 \, x^{2} - 10 \, x - 1\right )} \sqrt {x^{2} + x + 1} - \frac {7}{16} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]
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Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int \frac {x^3}{\sqrt {1+x+x^2}} \, dx=\left (\frac {x^{2}}{3} - \frac {5 x}{12} - \frac {1}{24}\right ) \sqrt {x^{2} + x + 1} + \frac {7 \operatorname {asinh}{\left (\frac {2 \sqrt {3} \left (x + \frac {1}{2}\right )}{3} \right )}}{16} \]
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Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{\sqrt {1+x+x^2}} \, dx=\frac {1}{3} \, \sqrt {x^{2} + x + 1} x^{2} - \frac {5}{12} \, \sqrt {x^{2} + x + 1} x - \frac {1}{24} \, \sqrt {x^{2} + x + 1} + \frac {7}{16} \, \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) \]
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Time = 0.33 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.74 \[ \int \frac {x^3}{\sqrt {1+x+x^2}} \, dx=\frac {1}{24} \, {\left (2 \, {\left (4 \, x - 5\right )} x - 1\right )} \sqrt {x^{2} + x + 1} - \frac {7}{16} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]
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Timed out. \[ \int \frac {x^3}{\sqrt {1+x+x^2}} \, dx=\int \frac {x^3}{\sqrt {x^2+x+1}} \,d x \]
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