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]

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Rubi [A]  time = 0.0095382, 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, Rules used = {619, 215} \[ \frac{\sinh ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[2 + 4*x + 3*x^2],x]

[Out]

ArcSinh[(2 + 3*x)/Sqrt[2]]/Sqrt[3]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{2+4 x+3 x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{8}}} \, dx,x,4+6 x\right )}{2 \sqrt{6}}\\ &=\frac{\sinh ^{-1}\left (\frac{2+3 x}{\sqrt{2}}\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.006345, 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]

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Maple [A]  time = 0.048, size = 15, normalized size = 0.8 \begin{align*}{\frac{\sqrt{3}}{3}{\it Arcsinh} \left ({\frac{3\,\sqrt{2}}{2} \left ( x+{\frac{2}{3}} \right ) } \right ) } \end{align*}

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))

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Maxima [A]  time = 1.76543, size = 22, normalized size = 1.22 \begin{align*} \frac{1}{3} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.91206, size = 105, normalized size = 5.83 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 4 \, x + 2}{\left (3 \, x + 2\right )} - 9 \, x^{2} - 12 \, x - 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)

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Giac [B]  time = 1.29968, size = 45, normalized size = 2.5 \begin{align*} -\frac{1}{3} \, \sqrt{3} \log \left (-\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 4 \, x + 2}\right )} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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