3.2391 \(\int \frac{3+2 x}{\sqrt{5+2 x+x^2}} \, dx\)

Optimal. Leaf size=23 \[ 2 \sqrt{x^2+2 x+5}+\sinh ^{-1}\left (\frac{x+1}{2}\right ) \]

[Out]

2*Sqrt[5 + 2*x + x^2] + ArcSinh[(1 + x)/2]

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Rubi [A]  time = 0.0330411, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ 2 \sqrt{x^2+2 x+5}+\sinh ^{-1}\left (\frac{x+1}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(3 + 2*x)/Sqrt[5 + 2*x + x^2],x]

[Out]

2*Sqrt[5 + 2*x + x^2] + ArcSinh[(1 + x)/2]

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Rubi in Sympy [A]  time = 6.57855, size = 32, normalized size = 1.39 \[ 2 \sqrt{x^{2} + 2 x + 5} + \operatorname{atanh}{\left (\frac{2 x + 2}{2 \sqrt{x^{2} + 2 x + 5}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+2*x)/(x**2+2*x+5)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0238528, size = 23, normalized size = 1. \[ 2 \sqrt{x^2+2 x+5}+\sinh ^{-1}\left (\frac{x+1}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 2*x)/Sqrt[5 + 2*x + x^2],x]

[Out]

2*Sqrt[5 + 2*x + x^2] + ArcSinh[(1 + x)/2]

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Maple [A]  time = 0.01, size = 20, normalized size = 0.9 \[{\it Arcsinh} \left ({\frac{1}{2}}+{\frac{x}{2}} \right ) +2\,\sqrt{{x}^{2}+2\,x+5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x+3)/(x^2+2*x+5)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.754895, size = 26, normalized size = 1.13 \[ 2 \, \sqrt{x^{2} + 2 \, x + 5} + \operatorname{arsinh}\left (\frac{1}{2} \, x + \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 3)/sqrt(x^2 + 2*x + 5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.227399, size = 105, normalized size = 4.57 \[ -\frac{2 \, x^{2} +{\left (x - \sqrt{x^{2} + 2 \, x + 5} + 1\right )} \log \left (-x + \sqrt{x^{2} + 2 \, x + 5} - 1\right ) - \sqrt{x^{2} + 2 \, x + 5}{\left (2 \, x + 1\right )} + 3 \, x + 9}{x - \sqrt{x^{2} + 2 \, x + 5} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 3)/sqrt(x^2 + 2*x + 5),x, algorithm="fricas")

[Out]

-(2*x^2 + (x - sqrt(x^2 + 2*x + 5) + 1)*log(-x + sqrt(x^2 + 2*x + 5) - 1) - sqrt
(x^2 + 2*x + 5)*(2*x + 1) + 3*x + 9)/(x - sqrt(x^2 + 2*x + 5) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 x + 3}{\sqrt{x^{2} + 2 x + 5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3+2*x)/(x**2+2*x+5)**(1/2),x)

[Out]

Integral((2*x + 3)/sqrt(x**2 + 2*x + 5), x)

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GIAC/XCAS [A]  time = 0.20815, size = 42, normalized size = 1.83 \[ 2 \, \sqrt{x^{2} + 2 \, x + 5} -{\rm ln}\left (-x + \sqrt{x^{2} + 2 \, x + 5} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x + 3)/sqrt(x^2 + 2*x + 5),x, algorithm="giac")

[Out]

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