Optimal. Leaf size=18 \[ \sqrt {9+x^2}+2 \sinh ^{-1}\left (\frac {x}{3}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {655, 221}
\begin {gather*} \sqrt {x^2+9}+2 \sinh ^{-1}\left (\frac {x}{3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 655
Rubi steps
\begin {align*} \int \frac {2+x}{\sqrt {9+x^2}} \, dx &=\sqrt {9+x^2}+2 \int \frac {1}{\sqrt {9+x^2}} \, dx\\ &=\sqrt {9+x^2}+2 \sinh ^{-1}\left (\frac {x}{3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 24, normalized size = 1.33 \begin {gather*} \sqrt {9+x^2}+2 \tanh ^{-1}\left (\frac {x}{\sqrt {9+x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 15, normalized size = 0.83
method | result | size |
default | \(2 \arcsinh \left (\frac {x}{3}\right )+\sqrt {x^{2}+9}\) | \(15\) |
risch | \(2 \arcsinh \left (\frac {x}{3}\right )+\sqrt {x^{2}+9}\) | \(15\) |
trager | \(\sqrt {x^{2}+9}-2 \ln \left (x -\sqrt {x^{2}+9}\right )\) | \(23\) |
meijerg | \(2 \arcsinh \left (\frac {x}{3}\right )+\frac {-3 \sqrt {\pi }+3 \sqrt {\pi }\, \sqrt {\frac {x^{2}}{9}+1}}{\sqrt {\pi }}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 14, normalized size = 0.78 \begin {gather*} \sqrt {x^{2} + 9} + 2 \, \operatorname {arsinh}\left (\frac {1}{3} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.67, size = 22, normalized size = 1.22 \begin {gather*} \sqrt {x^{2} + 9} - 2 \, \log \left (-x + \sqrt {x^{2} + 9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 14, normalized size = 0.78 \begin {gather*} \sqrt {x^{2} + 9} + 2 \operatorname {asinh}{\left (\frac {x}{3} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.82, size = 22, normalized size = 1.22 \begin {gather*} \sqrt {x^{2} + 9} - 2 \, \log \left (-x + \sqrt {x^{2} + 9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 14, normalized size = 0.78 \begin {gather*} 2\,\mathrm {asinh}\left (\frac {x}{3}\right )+\sqrt {x^2+9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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