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

Optimal. Leaf size=32 \[ \frac{\tanh ^{-1}\left (\frac{3 x+2}{\sqrt{3} \sqrt{3 x^2+4 x-2}}\right )}{\sqrt{3}} \]

[Out]

ArcTanh[(2 + 3*x)/(Sqrt[3]*Sqrt[-2 + 4*x + 3*x^2])]/Sqrt[3]

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Rubi [A]  time = 0.0077246, antiderivative size = 32, 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 = {621, 206} \[ \frac{\tanh ^{-1}\left (\frac{3 x+2}{\sqrt{3} \sqrt{3 x^2+4 x-2}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 + 4*x + 3*x^2],x]

[Out]

ArcTanh[(2 + 3*x)/(Sqrt[3]*Sqrt[-2 + 4*x + 3*x^2])]/Sqrt[3]

Rule 621

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-2+4 x+3 x^2}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{4+6 x}{\sqrt{-2+4 x+3 x^2}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{2+3 x}{\sqrt{3} \sqrt{-2+4 x+3 x^2}}\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0067269, size = 26, normalized size = 0.81 \[ \frac{\log \left (\sqrt{9 x^2+12 x-6}+3 x+2\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-2 + 4*x + 3*x^2],x]

[Out]

Log[2 + 3*x + Sqrt[-6 + 12*x + 9*x^2]]/Sqrt[3]

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Maple [A]  time = 0.049, size = 30, normalized size = 0.9 \begin{align*}{\frac{\sqrt{3}}{3}\ln \left ({\frac{ \left ( 2+3\,x \right ) \sqrt{3}}{3}}+\sqrt{3\,{x}^{2}+4\,x-2} \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*ln(1/3*(2+3*x)*3^(1/2)+(3*x^2+4*x-2)^(1/2))*3^(1/2)

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Maxima [A]  time = 1.74095, size = 38, normalized size = 1.19 \begin{align*} \frac{1}{3} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 4 \, x - 2} + 6 \, x + 4\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)*log(2*sqrt(3)*sqrt(3*x^2 + 4*x - 2) + 6*x + 4)

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Fricas [A]  time = 1.76923, size = 104, normalized size = 3.25 \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 - 1\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 - 1)

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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 [A]  time = 1.2625, size = 46, normalized size = 1.44 \begin{align*} -\frac{1}{3} \, \sqrt{3} \log \left ({\left | -\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 4 \, x - 2}\right )} - 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="giac")

[Out]

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