3.288 \(\int \frac{1}{\sqrt{-5+12 x+9 x^2}} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{3} \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{9 x^2+12 x-5}}\right ) \]

[Out]

ArcTanh[(2 + 3*x)/Sqrt[-5 + 12*x + 9*x^2]]/3

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Rubi [A]  time = 0.0054843, antiderivative size = 25, 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} \[ \frac{1}{3} \tanh ^{-1}\left (\frac{3 x+2}{\sqrt{9 x^2+12 x-5}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-5 + 12*x + 9*x^2],x]

[Out]

ArcTanh[(2 + 3*x)/Sqrt[-5 + 12*x + 9*x^2]]/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{-5+12 x+9 x^2}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{36-x^2} \, dx,x,\frac{12+18 x}{\sqrt{-5+12 x+9 x^2}}\right )\\ &=\frac{1}{3} \tanh ^{-1}\left (\frac{2+3 x}{\sqrt{-5+12 x+9 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0058602, size = 24, normalized size = 0.96 \[ \frac{1}{3} \log \left (\sqrt{9 x^2+12 x-5}+3 x+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-5 + 12*x + 9*x^2],x]

[Out]

Log[2 + 3*x + Sqrt[-5 + 12*x + 9*x^2]]/3

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Maple [A]  time = 0.004, size = 30, normalized size = 1.2 \begin{align*}{\frac{\sqrt{9}}{9}\ln \left ({\frac{ \left ( 6+9\,x \right ) \sqrt{9}}{9}}+\sqrt{9\,{x}^{2}+12\,x-5} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(9*x^2+12*x-5)^(1/2),x)

[Out]

1/9*ln(1/9*(6+9*x)*9^(1/2)+(9*x^2+12*x-5)^(1/2))*9^(1/2)

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Maxima [A]  time = 1.4183, size = 30, normalized size = 1.2 \begin{align*} \frac{1}{3} \, \log \left (18 \, x + 6 \, \sqrt{9 \, x^{2} + 12 \, x - 5} + 12\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+12*x-5)^(1/2),x, algorithm="maxima")

[Out]

1/3*log(18*x + 6*sqrt(9*x^2 + 12*x - 5) + 12)

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Fricas [A]  time = 1.98372, size = 61, normalized size = 2.44 \begin{align*} -\frac{1}{3} \, \log \left (-3 \, x + \sqrt{9 \, x^{2} + 12 \, x - 5} - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+12*x-5)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{9 x^{2} + 12 x - 5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x**2+12*x-5)**(1/2),x)

[Out]

Integral(1/sqrt(9*x**2 + 12*x - 5), x)

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Giac [A]  time = 1.07072, size = 28, normalized size = 1.12 \begin{align*} -\frac{1}{3} \, \log \left ({\left | -3 \, x + \sqrt{9 \, x^{2} + 12 \, x - 5} - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+12*x-5)^(1/2),x, algorithm="giac")

[Out]

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