3.13 \(\int \sqrt{4 x+x^2} \, dx\)

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]

((2 + x)*Sqrt[4*x + x^2])/2 - 4*ArcTanh[x/Sqrt[4*x + x^2]]

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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]

((2 + x)*Sqrt[4*x + x^2])/2 - 4*ArcTanh[x/Sqrt[4*x + x^2]]

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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]

(2*x + 4)*sqrt(x**2 + 4*x)/4 - 4*atanh(x/sqrt(x**2 + 4*x))

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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]

(Sqrt[x*(4 + x)]*(2 + x - (8*ArcSinh[Sqrt[x]/2])/(Sqrt[x]*Sqrt[4 + x])))/2

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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]

1/4*(2*x+4)*(x^2+4*x)^(1/2)-2*ln(2+x+(x^2+4*x)^(1/2))

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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]

1/2*sqrt(x^2 + 4*x)*x + sqrt(x^2 + 4*x) - 2*log(2*x + 2*sqrt(x^2 + 4*x) + 4)

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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]

-1/2*(x^4 + 8*x^3 + 19*x^2 - 4*(x^2 - sqrt(x^2 + 4*x)*(x + 2) + 4*x + 2)*log(-x
+ sqrt(x^2 + 4*x) - 2) - (x^3 + 6*x^2 + 9*x + 2)*sqrt(x^2 + 4*x) + 12*x - 2)/(x^
2 - sqrt(x^2 + 4*x)*(x + 2) + 4*x + 2)

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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]

Integral(sqrt(x**2 + 4*x), x)

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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]

1/2*sqrt(x^2 + 4*x)*(x + 2) + 2*ln(abs(-x + sqrt(x^2 + 4*x) - 2))