Optimal. Leaf size=35 \[ \frac {1}{2} (x+2) \sqrt {x^2+4 x}-4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+4 x}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {612, 620, 206} \[ \frac {1}{2} (x+2) \sqrt {x^2+4 x}-4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+4 x}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rubi steps
\begin {align*} \int \sqrt {4 x+x^2} \, dx &=\frac {1}{2} (2+x) \sqrt {4 x+x^2}-2 \int \frac {1}{\sqrt {4 x+x^2}} \, dx\\ &=\frac {1}{2} (2+x) \sqrt {4 x+x^2}-4 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {4 x+x^2}}\right )\\ &=\frac {1}{2} (2+x) \sqrt {4 x+x^2}-4 \tanh ^{-1}\left (\frac {x}{\sqrt {4 x+x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 40, normalized size = 1.14 \[ \frac {1}{2} \sqrt {x (x+4)} \left (x-\frac {8 \sinh ^{-1}\left (\frac {\sqrt {x}}{2}\right )}{\sqrt {x+4} \sqrt {x}}+2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 32, normalized size = 0.91 \[ \frac {1}{2} \, \sqrt {x^{2} + 4 \, x} {\left (x + 2\right )} + 2 \, \log \left (-x + \sqrt {x^{2} + 4 \, x} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 33, normalized size = 0.94 \[ \frac {1}{2} \, \sqrt {x^{2} + 4 \, x} {\left (x + 2\right )} + 2 \, \log \left ({\left | -x + \sqrt {x^{2} + 4 \, x} - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 33, normalized size = 0.94 \[ -2 \ln \left (x +2+\sqrt {x^{2}+4 x}\right )+\frac {\left (2 x +4\right ) \sqrt {x^{2}+4 x}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 41, normalized size = 1.17 \[ \frac {1}{2} \, \sqrt {x^{2} + 4 \, x} x + \sqrt {x^{2} + 4 \, x} - 2 \, \log \left (2 \, x + 2 \, \sqrt {x^{2} + 4 \, x} + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 29, normalized size = 0.83 \[ \sqrt {x^2+4\,x}\,\left (\frac {x}{2}+1\right )-2\,\ln \left (x+\sqrt {x\,\left (x+4\right )}+2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x^{2} + 4 x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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