3.152 \(\int \sqrt{5+x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{2} \sqrt{x^2+5} x+\frac{5}{2} \sinh ^{-1}\left (\frac{x}{\sqrt{5}}\right ) \]

[Out]

(x*Sqrt[5 + x^2])/2 + (5*ArcSinh[x/Sqrt[5]])/2

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Rubi [A]  time = 0.0100395, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{2} \sqrt{x^2+5} x+\frac{5}{2} \sinh ^{-1}\left (\frac{x}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[5 + x^2],x]

[Out]

(x*Sqrt[5 + x^2])/2 + (5*ArcSinh[x/Sqrt[5]])/2

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Rubi in Sympy [A]  time = 0.557144, size = 24, normalized size = 0.89 \[ \frac{x \sqrt{x^{2} + 5}}{2} + \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{5} x}{5} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+5)**(1/2),x)

[Out]

x*sqrt(x**2 + 5)/2 + 5*asinh(sqrt(5)*x/5)/2

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Mathematica [A]  time = 0.0129843, size = 27, normalized size = 1. \[ \frac{1}{2} \sqrt{x^2+5} x+\frac{5}{2} \sinh ^{-1}\left (\frac{x}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[5 + x^2],x]

[Out]

(x*Sqrt[5 + x^2])/2 + (5*ArcSinh[x/Sqrt[5]])/2

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Maple [A]  time = 0.004, size = 21, normalized size = 0.8 \[{\frac{5}{2}{\it Arcsinh} \left ({\frac{x\sqrt{5}}{5}} \right ) }+{\frac{x}{2}\sqrt{{x}^{2}+5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+5)^(1/2),x)

[Out]

5/2*arcsinh(1/5*x*5^(1/2))+1/2*x*(x^2+5)^(1/2)

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Maxima [A]  time = 1.50438, size = 27, normalized size = 1. \[ \frac{1}{2} \, \sqrt{x^{2} + 5} x + \frac{5}{2} \, \operatorname{arsinh}\left (\frac{1}{5} \, \sqrt{5} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + 5),x, algorithm="maxima")

[Out]

1/2*sqrt(x^2 + 5)*x + 5/2*arcsinh(1/5*sqrt(5)*x)

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Fricas [A]  time = 0.209638, size = 109, normalized size = 4.04 \[ -\frac{2 \, x^{4} + 10 \, x^{2} + 5 \,{\left (2 \, x^{2} - 2 \, \sqrt{x^{2} + 5} x + 5\right )} \log \left (-x + \sqrt{x^{2} + 5}\right ) -{\left (2 \, x^{3} + 5 \, x\right )} \sqrt{x^{2} + 5}}{2 \,{\left (2 \, x^{2} - 2 \, \sqrt{x^{2} + 5} x + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + 5),x, algorithm="fricas")

[Out]

-1/2*(2*x^4 + 10*x^2 + 5*(2*x^2 - 2*sqrt(x^2 + 5)*x + 5)*log(-x + sqrt(x^2 + 5))
- (2*x^3 + 5*x)*sqrt(x^2 + 5))/(2*x^2 - 2*sqrt(x^2 + 5)*x + 5)

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Sympy [A]  time = 0.271315, size = 24, normalized size = 0.89 \[ \frac{x \sqrt{x^{2} + 5}}{2} + \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{5} x}{5} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+5)**(1/2),x)

[Out]

x*sqrt(x**2 + 5)/2 + 5*asinh(sqrt(5)*x/5)/2

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GIAC/XCAS [A]  time = 0.224888, size = 34, normalized size = 1.26 \[ \frac{1}{2} \, \sqrt{x^{2} + 5} x - \frac{5}{2} \,{\rm ln}\left (-x + \sqrt{x^{2} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + 5),x, algorithm="giac")

[Out]

1/2*sqrt(x^2 + 5)*x - 5/2*ln(-x + sqrt(x^2 + 5))