Integrand size = 13, antiderivative size = 60 \[ \int x \sqrt {4+x^2} \log (x) \, dx=-\frac {4}{3} \sqrt {4+x^2}-\frac {1}{9} \left (4+x^2\right )^{3/2}+\frac {8}{3} \text {arctanh}\left (\frac {\sqrt {4+x^2}}{2}\right )+\frac {1}{3} \left (4+x^2\right )^{3/2} \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2376, 272, 52, 65, 213} \[ \int x \sqrt {4+x^2} \log (x) \, dx=\frac {8}{3} \text {arctanh}\left (\frac {\sqrt {x^2+4}}{2}\right )-\frac {1}{9} \left (x^2+4\right )^{3/2}-\frac {4 \sqrt {x^2+4}}{3}+\frac {1}{3} \left (x^2+4\right )^{3/2} \log (x) \]
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Rule 52
Rule 65
Rule 213
Rule 272
Rule 2376
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \left (4+x^2\right )^{3/2} \log (x)-\frac {1}{3} \int \frac {\left (4+x^2\right )^{3/2}}{x} \, dx \\ & = \frac {1}{3} \left (4+x^2\right )^{3/2} \log (x)-\frac {1}{6} \text {Subst}\left (\int \frac {(4+x)^{3/2}}{x} \, dx,x,x^2\right ) \\ & = -\frac {1}{9} \left (4+x^2\right )^{3/2}+\frac {1}{3} \left (4+x^2\right )^{3/2} \log (x)-\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {4+x}}{x} \, dx,x,x^2\right ) \\ & = -\frac {4}{3} \sqrt {4+x^2}-\frac {1}{9} \left (4+x^2\right )^{3/2}+\frac {1}{3} \left (4+x^2\right )^{3/2} \log (x)-\frac {8}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {4+x}} \, dx,x,x^2\right ) \\ & = -\frac {4}{3} \sqrt {4+x^2}-\frac {1}{9} \left (4+x^2\right )^{3/2}+\frac {1}{3} \left (4+x^2\right )^{3/2} \log (x)-\frac {16}{3} \text {Subst}\left (\int \frac {1}{-4+x^2} \, dx,x,\sqrt {4+x^2}\right ) \\ & = -\frac {4}{3} \sqrt {4+x^2}-\frac {1}{9} \left (4+x^2\right )^{3/2}+\frac {8}{3} \tanh ^{-1}\left (\frac {\sqrt {4+x^2}}{2}\right )+\frac {1}{3} \left (4+x^2\right )^{3/2} \log (x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.88 \[ \int x \sqrt {4+x^2} \log (x) \, dx=\frac {1}{3} \left (-\frac {1}{3} \sqrt {4+x^2} \left (16+x^2\right )-8 \log (x)+\left (4+x^2\right )^{3/2} \log (x)+8 \log \left (2+\sqrt {4+x^2}\right )\right ) \]
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Time = 0.44 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.25
method | result | size |
meijerg | \(\left (-\frac {2 \sqrt {1+\frac {x^{2}}{4}}}{9}+\frac {2 \ln \left (x \right ) \sqrt {1+\frac {x^{2}}{4}}}{3}\right ) x^{2}+\frac {32}{9}-\frac {32 \sqrt {1+\frac {x^{2}}{4}}}{9}+\ln \left (x \right ) \left (-\frac {8}{3}+\frac {8 \sqrt {1+\frac {x^{2}}{4}}}{3}\right )+\frac {8 \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{2}}{4}}}{2}\right )}{3}\) | \(75\) |
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Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.90 \[ \int x \sqrt {4+x^2} \log (x) \, dx=-\frac {1}{9} \, {\left (x^{2} - 3 \, {\left (x^{2} + 4\right )} \log \left (x\right ) + 16\right )} \sqrt {x^{2} + 4} + \frac {8}{3} \, \log \left (-x + \sqrt {x^{2} + 4} + 2\right ) - \frac {8}{3} \, \log \left (-x + \sqrt {x^{2} + 4} - 2\right ) \]
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Time = 7.76 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.18 \[ \int x \sqrt {4+x^2} \log (x) \, dx=\left (\frac {x^{2}}{3} + \frac {4}{3}\right ) \sqrt {x^{2} + 4} \log {\left (x \right )} - \frac {\left (x^{2} + 4\right )^{\frac {3}{2}}}{9} - \frac {4 \sqrt {x^{2} + 4}}{3} - \frac {4 \log {\left (\sqrt {x^{2} + 4} - 2 \right )}}{3} + \frac {4 \log {\left (\sqrt {x^{2} + 4} + 2 \right )}}{3} \]
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Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.65 \[ \int x \sqrt {4+x^2} \log (x) \, dx=\frac {1}{3} \, {\left (x^{2} + 4\right )}^{\frac {3}{2}} \log \left (x\right ) - \frac {1}{9} \, {\left (x^{2} + 4\right )}^{\frac {3}{2}} - \frac {4}{3} \, \sqrt {x^{2} + 4} + \frac {8}{3} \, \operatorname {arsinh}\left (\frac {2}{{\left | x \right |}}\right ) \]
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Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.90 \[ \int x \sqrt {4+x^2} \log (x) \, dx=\frac {1}{3} \, {\left (x^{2} + 4\right )}^{\frac {3}{2}} \log \left (x\right ) - \frac {1}{9} \, {\left (x^{2} + 4\right )}^{\frac {3}{2}} - \frac {4}{3} \, \sqrt {x^{2} + 4} + \frac {4}{3} \, \log \left (\sqrt {x^{2} + 4} + 2\right ) - \frac {4}{3} \, \log \left (\sqrt {x^{2} + 4} - 2\right ) \]
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Timed out. \[ \int x \sqrt {4+x^2} \log (x) \, dx=\int x\,\ln \left (x\right )\,\sqrt {x^2+4} \,d x \]
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