Integrand size = 14, antiderivative size = 50 \[ \int x \sqrt {4+2 x+x^2} \, dx=-\frac {1}{2} (1+x) \sqrt {4+2 x+x^2}+\frac {1}{3} \left (4+2 x+x^2\right )^{3/2}-\frac {3}{2} \text {arcsinh}\left (\frac {1+x}{\sqrt {3}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 50, 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 = {654, 626, 633, 221} \[ \int x \sqrt {4+2 x+x^2} \, dx=-\frac {3}{2} \text {arcsinh}\left (\frac {x+1}{\sqrt {3}}\right )+\frac {1}{3} \left (x^2+2 x+4\right )^{3/2}-\frac {1}{2} (x+1) \sqrt {x^2+2 x+4} \]
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Rule 221
Rule 626
Rule 633
Rule 654
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \left (4+2 x+x^2\right )^{3/2}-\int \sqrt {4+2 x+x^2} \, dx \\ & = -\frac {1}{2} (1+x) \sqrt {4+2 x+x^2}+\frac {1}{3} \left (4+2 x+x^2\right )^{3/2}-\frac {3}{2} \int \frac {1}{\sqrt {4+2 x+x^2}} \, dx \\ & = -\frac {1}{2} (1+x) \sqrt {4+2 x+x^2}+\frac {1}{3} \left (4+2 x+x^2\right )^{3/2}-\frac {1}{4} \sqrt {3} \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{12}}} \, dx,x,2+2 x\right ) \\ & = -\frac {1}{2} (1+x) \sqrt {4+2 x+x^2}+\frac {1}{3} \left (4+2 x+x^2\right )^{3/2}-\frac {3}{2} \sinh ^{-1}\left (\frac {1+x}{\sqrt {3}}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94 \[ \int x \sqrt {4+2 x+x^2} \, dx=\frac {1}{6} \sqrt {4+2 x+x^2} \left (5+x+2 x^2\right )+\frac {3}{2} \log \left (-1-x+\sqrt {4+2 x+x^2}\right ) \]
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Time = 0.38 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.66
method | result | size |
risch | \(\frac {\left (2 x^{2}+x +5\right ) \sqrt {x^{2}+2 x +4}}{6}-\frac {3 \,\operatorname {arcsinh}\left (\frac {\left (1+x \right ) \sqrt {3}}{3}\right )}{2}\) | \(33\) |
trager | \(\left (\frac {1}{3} x^{2}+\frac {1}{6} x +\frac {5}{6}\right ) \sqrt {x^{2}+2 x +4}-\frac {3 \ln \left (1+x +\sqrt {x^{2}+2 x +4}\right )}{2}\) | \(39\) |
default | \(\frac {\left (x^{2}+2 x +4\right )^{\frac {3}{2}}}{3}-\frac {\left (2 x +2\right ) \sqrt {x^{2}+2 x +4}}{4}-\frac {3 \,\operatorname {arcsinh}\left (\frac {\left (1+x \right ) \sqrt {3}}{3}\right )}{2}\) | \(42\) |
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Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.78 \[ \int x \sqrt {4+2 x+x^2} \, dx=\frac {1}{6} \, {\left (2 \, x^{2} + x + 5\right )} \sqrt {x^{2} + 2 \, x + 4} + \frac {3}{2} \, \log \left (-x + \sqrt {x^{2} + 2 \, x + 4} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.74 \[ \int x \sqrt {4+2 x+x^2} \, dx=\left (\frac {x^{2}}{3} + \frac {x}{6} + \frac {5}{6}\right ) \sqrt {x^{2} + 2 x + 4} - \frac {3 \operatorname {asinh}{\left (\frac {\sqrt {3} \left (x + 1\right )}{3} \right )}}{2} \]
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Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98 \[ \int x \sqrt {4+2 x+x^2} \, dx=\frac {1}{3} \, {\left (x^{2} + 2 \, x + 4\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {x^{2} + 2 \, x + 4} x - \frac {1}{2} \, \sqrt {x^{2} + 2 \, x + 4} - \frac {3}{2} \, \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.80 \[ \int x \sqrt {4+2 x+x^2} \, dx=\frac {1}{6} \, {\left ({\left (2 \, x + 1\right )} x + 5\right )} \sqrt {x^{2} + 2 \, x + 4} + \frac {3}{2} \, \log \left (-x + \sqrt {x^{2} + 2 \, x + 4} - 1\right ) \]
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Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.78 \[ \int x \sqrt {4+2 x+x^2} \, dx=\frac {\sqrt {x^2+2\,x+4}\,\left (8\,x^2+4\,x+20\right )}{24}-\frac {3\,\ln \left (x+\sqrt {x^2+2\,x+4}+1\right )}{2} \]
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