3.117 \(\int \frac{1}{\sqrt{2+4 x+3 x^2}} \, dx\)

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

[Out]

ArcSinh[(2 + 3*x)/Sqrt[2]]/Sqrt[3]

_______________________________________________________________________________________

Rubi [A]  time = 0.022426, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[2 + 4*x + 3*x^2],x]

[Out]

ArcSinh[(2 + 3*x)/Sqrt[2]]/Sqrt[3]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.48068, size = 32, normalized size = 1.78 \[ \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 4\right )}{6 \sqrt{3 x^{2} + 4 x + 2}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*x**2+4*x+2)**(1/2),x)

[Out]

sqrt(3)*atanh(sqrt(3)*(6*x + 4)/(6*sqrt(3*x**2 + 4*x + 2)))/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.01124, size = 18, normalized size = 1. \[ \frac{\sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[2 + 4*x + 3*x^2],x]

[Out]

ArcSinh[(2 + 3*x)/Sqrt[2]]/Sqrt[3]

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 15, normalized size = 0.8 \[{\frac{\sqrt{3}}{3}{\it Arcsinh} \left ({\frac{3\,\sqrt{2}}{2} \left ( x+{\frac{2}{3}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^2+4*x+2)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.828608, size = 22, normalized size = 1.22 \[ \frac{1}{3} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^2 + 4*x + 2),x, algorithm="maxima")

[Out]

1/3*sqrt(3)*arcsinh(1/2*sqrt(2)*(3*x + 2))

_______________________________________________________________________________________

Fricas [A]  time = 0.231959, size = 55, normalized size = 3.06 \[ \frac{1}{6} \, \sqrt{3} \log \left (-\sqrt{3}{\left (9 \, x^{2} + 12 \, x + 5\right )} - 3 \, \sqrt{3 \, x^{2} + 4 \, x + 2}{\left (3 \, x + 2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^2 + 4*x + 2),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log(-sqrt(3)*(9*x^2 + 12*x + 5) - 3*sqrt(3*x^2 + 4*x + 2)*(3*x + 2))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 x^{2} + 4 x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*x**2+4*x+2)**(1/2),x)

[Out]

Integral(1/sqrt(3*x**2 + 4*x + 2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214246, size = 45, normalized size = 2.5 \[ -\frac{1}{3} \, \sqrt{3}{\rm ln}\left (-\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 4 \, x + 2}\right )} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^2 + 4*x + 2),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*ln(-sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 4*x + 2)) - 2)