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