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]

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

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \[ \frac {\sinh ^{-1}\left (\frac {3 x+2}{\sqrt {2}}\right )}{\sqrt {3}} \]

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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fricas [B]  time = 0.96, size = 38, normalized size = 2.11 \[ \frac {1}{6} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 4 \, x + 2} {\left (3 \, x + 2\right )} - 9 \, x^{2} - 12 \, x - 5\right ) \]

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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giac [B]  time = 0.62, size = 33, normalized size = 1.83 \[ -\frac {1}{3} \, \sqrt {3} \log \left (-\sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 4 \, x + 2}\right )} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)

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maple [A]  time = 0.05, size = 15, normalized size = 0.83 \[ \frac {\sqrt {3}\, \arcsinh \left (\frac {3 \sqrt {2}\, \left (x +\frac {2}{3}\right )}{2}\right )}{3} \]

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 = 2.88, size = 16, normalized size = 0.89 \[ \frac {1}{3} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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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mupad [B]  time = 0.22, size = 26, normalized size = 1.44 \[ \frac {\sqrt {3}\,\ln \left (\sqrt {3}\,\left (x+\frac {2}{3}\right )+\sqrt {3\,x^2+4\,x+2}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x + 3*x^2 + 2)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {3 x^{2} + 4 x + 2}}\, dx \]

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)

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