Integrand size = 11, antiderivative size = 48 \[ \int x \sqrt {x+x^2} \, dx=-\frac {1}{8} (1+2 x) \sqrt {x+x^2}+\frac {1}{3} \left (x+x^2\right )^{3/2}+\frac {1}{8} \text {arctanh}\left (\frac {x}{\sqrt {x+x^2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {654, 626, 634, 212} \[ \int x \sqrt {x+x^2} \, dx=\frac {1}{8} \text {arctanh}\left (\frac {x}{\sqrt {x^2+x}}\right )+\frac {1}{3} \left (x^2+x\right )^{3/2}-\frac {1}{8} (2 x+1) \sqrt {x^2+x} \]
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Rule 212
Rule 626
Rule 634
Rule 654
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \left (x+x^2\right )^{3/2}-\frac {1}{2} \int \sqrt {x+x^2} \, dx \\ & = -\frac {1}{8} (1+2 x) \sqrt {x+x^2}+\frac {1}{3} \left (x+x^2\right )^{3/2}+\frac {1}{16} \int \frac {1}{\sqrt {x+x^2}} \, dx \\ & = -\frac {1}{8} (1+2 x) \sqrt {x+x^2}+\frac {1}{3} \left (x+x^2\right )^{3/2}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {x+x^2}}\right ) \\ & = -\frac {1}{8} (1+2 x) \sqrt {x+x^2}+\frac {1}{3} \left (x+x^2\right )^{3/2}+\frac {1}{8} \tanh ^{-1}\left (\frac {x}{\sqrt {x+x^2}}\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.10 \[ \int x \sqrt {x+x^2} \, dx=\frac {1}{24} \sqrt {x (1+x)} \left (-3+2 x+8 x^2-\frac {3 \log \left (-\sqrt {x}+\sqrt {1+x}\right )}{\sqrt {x} \sqrt {1+x}}\right ) \]
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Time = 2.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.77
method | result | size |
trager | \(\left (\frac {1}{3} x^{2}+\frac {1}{12} x -\frac {1}{8}\right ) \sqrt {x^{2}+x}-\frac {\ln \left (2 \sqrt {x^{2}+x}-1-2 x \right )}{16}\) | \(37\) |
default | \(\frac {\left (x^{2}+x \right )^{\frac {3}{2}}}{3}-\frac {\left (1+2 x \right ) \sqrt {x^{2}+x}}{8}+\frac {\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{16}\) | \(38\) |
risch | \(\frac {\left (8 x^{2}+2 x -3\right ) x \left (1+x \right )}{24 \sqrt {x \left (1+x \right )}}+\frac {\ln \left (x +\frac {1}{2}+\sqrt {x^{2}+x}\right )}{16}\) | \(38\) |
meijerg | \(-\frac {\frac {\sqrt {\pi }\, \sqrt {x}\, \left (-40 x^{2}-10 x +15\right ) \sqrt {1+x}}{60}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\sqrt {x}\right )}{4}}{2 \sqrt {\pi }}\) | \(39\) |
pseudoelliptic | \(-\frac {x^{3} \left (-16 \sqrt {x \left (1+x \right )}\, x^{2}-4 x \sqrt {x \left (1+x \right )}+3 \ln \left (\frac {\sqrt {x \left (1+x \right )}-x}{x}\right )-3 \ln \left (\frac {x +\sqrt {x \left (1+x \right )}}{x}\right )+6 \sqrt {x \left (1+x \right )}\right )}{48 \left (\sqrt {x \left (1+x \right )}-x \right )^{3} \left (x +\sqrt {x \left (1+x \right )}\right )^{3}}\) | \(96\) |
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Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.77 \[ \int x \sqrt {x+x^2} \, dx=\frac {1}{24} \, {\left (8 \, x^{2} + 2 \, x - 3\right )} \sqrt {x^{2} + x} - \frac {1}{16} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x} - 1\right ) \]
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Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.77 \[ \int x \sqrt {x+x^2} \, dx=\sqrt {x^{2} + x} \left (\frac {x^{2}}{3} + \frac {x}{12} - \frac {1}{8}\right ) + \frac {\log {\left (2 x + 2 \sqrt {x^{2} + x} + 1 \right )}}{16} \]
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Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96 \[ \int x \sqrt {x+x^2} \, dx=\frac {1}{3} \, {\left (x^{2} + x\right )}^{\frac {3}{2}} - \frac {1}{4} \, \sqrt {x^{2} + x} x - \frac {1}{8} \, \sqrt {x^{2} + x} + \frac {1}{16} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} + x} + 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.79 \[ \int x \sqrt {x+x^2} \, dx=\frac {1}{24} \, {\left (2 \, {\left (4 \, x + 1\right )} x - 3\right )} \sqrt {x^{2} + x} - \frac {1}{16} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} + x} - 1 \right |}\right ) \]
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Time = 9.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.69 \[ \int x \sqrt {x+x^2} \, dx=\frac {\ln \left (x+\sqrt {x\,\left (x+1\right )}+\frac {1}{2}\right )}{16}+\frac {\sqrt {x^2+x}\,\left (8\,x^2+2\,x-3\right )}{24} \]
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