Optimal. Leaf size=27 \[ \frac {1}{2} x \sqrt {5+x^2}+\frac {5}{2} \sinh ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {201, 221}
\begin {gather*} \frac {1}{2} \sqrt {x^2+5} x+\frac {5}{2} \sinh ^{-1}\left (\frac {x}{\sqrt {5}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rubi steps
\begin {align*} \int \sqrt {5+x^2} \, dx &=\frac {1}{2} x \sqrt {5+x^2}+\frac {5}{2} \int \frac {1}{\sqrt {5+x^2}} \, dx\\ &=\frac {1}{2} x \sqrt {5+x^2}+\frac {5}{2} \sinh ^{-1}\left (\frac {x}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 31, normalized size = 1.15 \begin {gather*} \frac {1}{2} x \sqrt {5+x^2}+\frac {5}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {5+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 21, normalized size = 0.78
method | result | size |
default | \(\frac {5 \arcsinh \left (\frac {x \sqrt {5}}{5}\right )}{2}+\frac {x \sqrt {x^{2}+5}}{2}\) | \(21\) |
risch | \(\frac {5 \arcsinh \left (\frac {x \sqrt {5}}{5}\right )}{2}+\frac {x \sqrt {x^{2}+5}}{2}\) | \(21\) |
trager | \(\frac {x \sqrt {x^{2}+5}}{2}+\frac {5 \ln \left (x +\sqrt {x^{2}+5}\right )}{2}\) | \(24\) |
meijerg | \(-\frac {5 \left (-\frac {2 \sqrt {\pi }\, x \sqrt {5}\, \sqrt {1+\frac {x^{2}}{5}}}{5}-2 \sqrt {\pi }\, \arcsinh \left (\frac {x \sqrt {5}}{5}\right )\right )}{4 \sqrt {\pi }}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.43, size = 20, normalized size = 0.74 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 5} x + \frac {5}{2} \, \operatorname {arsinh}\left (\frac {1}{5} \, \sqrt {5} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.57, size = 25, normalized size = 0.93 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 5} x - \frac {5}{2} \, \log \left (-x + \sqrt {x^{2} + 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 24, normalized size = 0.89 \begin {gather*} \frac {x \sqrt {x^{2} + 5}}{2} + \frac {5 \operatorname {asinh}{\left (\frac {\sqrt {5} x}{5} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 25, normalized size = 0.93 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 5} x - \frac {5}{2} \, \log \left (-x + \sqrt {x^{2} + 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 20, normalized size = 0.74 \begin {gather*} \frac {5\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x}{5}\right )}{2}+\frac {x\,\sqrt {x^2+5}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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