Optimal. Leaf size=60 \[ -\frac {4}{3} \sqrt {4+x^2}-\frac {1}{9} \left (4+x^2\right )^{3/2}+\frac {8}{3} \tanh ^{-1}\left (\frac {\sqrt {4+x^2}}{2}\right )+\frac {1}{3} \left (4+x^2\right )^{3/2} \log (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2376, 272, 52,
65, 213} \begin {gather*} -\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)+\frac {8}{3} \tanh ^{-1}\left (\frac {\sqrt {x^2+4}}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 213
Rule 272
Rule 2376
Rubi steps
\begin {align*} \int x \sqrt {4+x^2} \log (x) \, dx &=\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*}
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Mathematica [A]
time = 0.03, size = 53, normalized size = 0.88 \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 75, normalized size = 1.25
method | result | size |
meijerg | \(\left (-\frac {2 \sqrt {1+\frac {x^{2}}{4}}}{9}+\frac {2 \ln \left (x \right ) \sqrt {1+\frac {x^{2}}{4}}}{3}\right ) x^{2}+\frac {32}{9}-\frac {32 \sqrt {1+\frac {x^{2}}{4}}}{9}+\ln \left (x \right ) \left (-\frac {8}{3}+\frac {8 \sqrt {1+\frac {x^{2}}{4}}}{3}\right )+\frac {8 \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{2}}{4}}}{2}\right )}{3}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 39, normalized size = 0.65 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 54, normalized size = 0.90 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 14.58, size = 65, normalized size = 1.08 \begin {gather*} \frac {\left (x^{2} + 4\right )^{\frac {3}{2}} \log {\left (x \right )}}{3} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.05, size = 54, normalized size = 0.90 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,\ln \left (x\right )\,\sqrt {x^2+4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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