Integrand size = 14, antiderivative size = 56 \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=-\frac {2 x^2 (2+x)}{3 \sqrt {1+x+x^2}}+\frac {1}{3} (5+2 x) \sqrt {1+x+x^2}-\frac {3}{2} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 56, 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 = {752, 793, 633, 221} \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=-\frac {3}{2} \text {arcsinh}\left (\frac {2 x+1}{\sqrt {3}}\right )-\frac {2 (x+2) x^2}{3 \sqrt {x^2+x+1}}+\frac {1}{3} (2 x+5) \sqrt {x^2+x+1} \]
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Rule 221
Rule 633
Rule 752
Rule 793
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^2 (2+x)}{3 \sqrt {1+x+x^2}}+\frac {2}{3} \int \frac {x (4+2 x)}{\sqrt {1+x+x^2}} \, dx \\ & = -\frac {2 x^2 (2+x)}{3 \sqrt {1+x+x^2}}+\frac {1}{3} (5+2 x) \sqrt {1+x+x^2}-\frac {3}{2} \int \frac {1}{\sqrt {1+x+x^2}} \, dx \\ & = -\frac {2 x^2 (2+x)}{3 \sqrt {1+x+x^2}}+\frac {1}{3} (5+2 x) \sqrt {1+x+x^2}-\frac {1}{2} \sqrt {3} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right ) \\ & = -\frac {2 x^2 (2+x)}{3 \sqrt {1+x+x^2}}+\frac {1}{3} (5+2 x) \sqrt {1+x+x^2}-\frac {3}{2} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84 \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=\frac {5+7 x+3 x^2}{3 \sqrt {1+x+x^2}}+\frac {3}{2} \log \left (-1-2 x+2 \sqrt {1+x+x^2}\right ) \]
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Time = 0.42 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.59
method | result | size |
risch | \(\frac {3 x^{2}+7 x +5}{3 \sqrt {x^{2}+x +1}}-\frac {3 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{2}\) | \(33\) |
trager | \(\frac {3 x^{2}+7 x +5}{3 \sqrt {x^{2}+x +1}}+\frac {3 \ln \left (2 \sqrt {x^{2}+x +1}-1-2 x \right )}{2}\) | \(40\) |
default | \(\frac {x^{2}}{\sqrt {x^{2}+x +1}}+\frac {3 x}{2 \sqrt {x^{2}+x +1}}+\frac {5}{4 \sqrt {x^{2}+x +1}}+\frac {\frac {5}{12}+\frac {5 x}{6}}{\sqrt {x^{2}+x +1}}-\frac {3 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{2}\) | \(61\) |
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Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.14 \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=\frac {19 \, x^{2} + 18 \, {\left (x^{2} + x + 1\right )} \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) + 4 \, {\left (3 \, x^{2} + 7 \, x + 5\right )} \sqrt {x^{2} + x + 1} + 19 \, x + 19}{12 \, {\left (x^{2} + x + 1\right )}} \]
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\[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=\int \frac {x^{3}}{\left (x^{2} + x + 1\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84 \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=\frac {x^{2}}{\sqrt {x^{2} + x + 1}} + \frac {7 \, x}{3 \, \sqrt {x^{2} + x + 1}} + \frac {5}{3 \, \sqrt {x^{2} + x + 1}} - \frac {3}{2} \, \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.68 \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=\frac {{\left (3 \, x + 7\right )} x + 5}{3 \, \sqrt {x^{2} + x + 1}} + \frac {3}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]
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Timed out. \[ \int \frac {x^3}{\left (1+x+x^2\right )^{3/2}} \, dx=\int \frac {x^3}{{\left (x^2+x+1\right )}^{3/2}} \,d x \]
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