Optimal. Leaf size=35 \[ \frac{1}{2} (x+2) \sqrt{x^2+4 x}-4 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2+4 x}}\right ) \]
[Out]
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Rubi [A] time = 0.0195852, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{1}{2} (x+2) \sqrt{x^2+4 x}-4 \tanh ^{-1}\left (\frac{x}{\sqrt{x^2+4 x}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[4*x + x^2],x]
[Out]
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Rubi in Sympy [A] time = 1.47816, size = 31, normalized size = 0.89 \[ \frac{\left (2 x + 4\right ) \sqrt{x^{2} + 4 x}}{4} - 4 \operatorname{atanh}{\left (\frac{x}{\sqrt{x^{2} + 4 x}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2+4*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0432707, 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.
[In] Integrate[Sqrt[4*x + x^2],x]
[Out]
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Maple [A] time = 0.005, size = 33, normalized size = 0.9 \[{\frac{2\,x+4}{4}\sqrt{{x}^{2}+4\,x}}-2\,\ln \left ( 2+x+\sqrt{{x}^{2}+4\,x} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2+4*x)^(1/2),x)
[Out]
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Maxima [A] time = 0.749732, size = 55, normalized size = 1.57 \[ \frac{1}{2} \, \sqrt{x^{2} + 4 \, x} x + \sqrt{x^{2} + 4 \, x} - 2 \, \log \left (2 \, x + 2 \, \sqrt{x^{2} + 4 \, x} + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 4*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231899, size = 144, normalized size = 4.11 \[ -\frac{x^{4} + 8 \, x^{3} + 19 \, x^{2} - 4 \,{\left (x^{2} - \sqrt{x^{2} + 4 \, x}{\left (x + 2\right )} + 4 \, x + 2\right )} \log \left (-x + \sqrt{x^{2} + 4 \, x} - 2\right ) -{\left (x^{3} + 6 \, x^{2} + 9 \, x + 2\right )} \sqrt{x^{2} + 4 \, x} + 12 \, x - 2}{2 \,{\left (x^{2} - \sqrt{x^{2} + 4 \, x}{\left (x + 2\right )} + 4 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 4*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{2} + 4 x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2+4*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210889, size = 45, normalized size = 1.29 \[ \frac{1}{2} \, \sqrt{x^{2} + 4 \, x}{\left (x + 2\right )} + 2 \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 4 \, x} - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2 + 4*x),x, algorithm="giac")
[Out]