Optimal. Leaf size=35 \[ \frac {1}{2} (2+x) \sqrt {4 x+x^2}-4 \tanh ^{-1}\left (\frac {x}{\sqrt {4 x+x^2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, 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 = {626, 634, 212}
\begin {gather*} \frac {1}{2} (x+2) \sqrt {x^2+4 x}-4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+4 x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 626
Rule 634
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 \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 42, normalized size = 1.20 \begin {gather*} \frac {1}{2} \sqrt {x (4+x)} \left (2+x-\frac {8 \tanh ^{-1}\left (\sqrt {\frac {x}{4+x}}\right )}{\sqrt {x} \sqrt {4+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.39, size = 33, normalized size = 0.94
method | result | size |
trager | \(\left (\frac {x}{2}+1\right ) \sqrt {x^{2}+4 x}-2 \ln \left (2+x +\sqrt {x^{2}+4 x}\right )\) | \(32\) |
default | \(\frac {\left (2 x +4\right ) \sqrt {x^{2}+4 x}}{4}-2 \ln \left (2+x +\sqrt {x^{2}+4 x}\right )\) | \(33\) |
risch | \(\frac {\left (2+x \right ) x \left (x +4\right )}{2 \sqrt {x \left (x +4\right )}}-2 \ln \left (2+x +\sqrt {x^{2}+4 x}\right )\) | \(33\) |
meijerg | \(-\frac {8 \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \left (\frac {3 x}{2}+3\right ) \sqrt {\frac {x}{4}+1}}{12}+\frac {\sqrt {\pi }\, \arcsinh \left (\frac {\sqrt {x}}{2}\right )}{2}\right )}{\sqrt {\pi }}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 41, normalized size = 1.17 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.72, size = 32, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} + 4 \, x} {\left (x + 2\right )} + 2 \, \log \left (-x + \sqrt {x^{2} + 4 \, x} - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{2} + 4 x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.54, size = 33, normalized size = 0.94 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.20, size = 29, normalized size = 0.83 \begin {gather*} \sqrt {x^2+4\,x}\,\left (\frac {x}{2}+1\right )-2\,\ln \left (x+\sqrt {x\,\left (x+4\right )}+2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________