3.576 \(\int \frac{2+x}{\sqrt{9+x^2}} \, dx\)

Optimal. Leaf size=18 \[ \sqrt{x^2+9}+2 \sinh ^{-1}\left (\frac{x}{3}\right ) \]

[Out]

Sqrt[9 + x^2] + 2*ArcSinh[x/3]

_______________________________________________________________________________________

Rubi [A]  time = 0.019318, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \sqrt{x^2+9}+2 \sinh ^{-1}\left (\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 + x)/Sqrt[9 + x^2],x]

[Out]

Sqrt[9 + x^2] + 2*ArcSinh[x/3]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.72365, size = 14, normalized size = 0.78 \[ \sqrt{x^{2} + 9} + 2 \operatorname{asinh}{\left (\frac{x}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+x)/(x**2+9)**(1/2),x)

[Out]

sqrt(x**2 + 9) + 2*asinh(x/3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0137404, size = 18, normalized size = 1. \[ \sqrt{x^2+9}+2 \sinh ^{-1}\left (\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + x)/Sqrt[9 + x^2],x]

[Out]

Sqrt[9 + x^2] + 2*ArcSinh[x/3]

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 15, normalized size = 0.8 \[ 2\,{\it Arcsinh} \left ( x/3 \right ) +\sqrt{{x}^{2}+9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+x)/(x^2+9)^(1/2),x)

[Out]

2*arcsinh(1/3*x)+(x^2+9)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.792654, size = 19, normalized size = 1.06 \[ \sqrt{x^{2} + 9} + 2 \, \operatorname{arsinh}\left (\frac{1}{3} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 2)/sqrt(x^2 + 9),x, algorithm="maxima")

[Out]

sqrt(x^2 + 9) + 2*arcsinh(1/3*x)

_______________________________________________________________________________________

Fricas [A]  time = 0.212925, size = 74, normalized size = 4.11 \[ -\frac{x^{2} + 2 \,{\left (x - \sqrt{x^{2} + 9}\right )} \log \left (-x + \sqrt{x^{2} + 9}\right ) - \sqrt{x^{2} + 9} x + 9}{x - \sqrt{x^{2} + 9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 2)/sqrt(x^2 + 9),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.326966, size = 14, normalized size = 0.78 \[ \sqrt{x^{2} + 9} + 2 \operatorname{asinh}{\left (\frac{x}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+x)/(x**2+9)**(1/2),x)

[Out]

sqrt(x**2 + 9) + 2*asinh(x/3)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213517, size = 30, normalized size = 1.67 \[ \sqrt{x^{2} + 9} - 2 \,{\rm ln}\left (-x + \sqrt{x^{2} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 2)/sqrt(x^2 + 9),x, algorithm="giac")

[Out]

sqrt(x^2 + 9) - 2*ln(-x + sqrt(x^2 + 9))