3.66 \(\int \frac{1}{\sqrt{4+12 x+9 x^2}} \, dx\)

Optimal. Leaf size=29 \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{9 x^2+12 x+4}} \]

[Out]

((2 + 3*x)*Log[2 + 3*x])/(3*Sqrt[4 + 12*x + 9*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0048338, antiderivative size = 29, 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 = {608, 31} \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{9 x^2+12 x+4}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[4 + 12*x + 9*x^2],x]

[Out]

((2 + 3*x)*Log[2 + 3*x])/(3*Sqrt[4 + 12*x + 9*x^2])

Rule 608

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{4+12 x+9 x^2}} \, dx &=\frac{(6+9 x) \int \frac{1}{6+9 x} \, dx}{\sqrt{4+12 x+9 x^2}}\\ &=\frac{(2+3 x) \log (2+3 x)}{3 \sqrt{4+12 x+9 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0063044, size = 26, normalized size = 0.9 \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{(3 x+2)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[4 + 12*x + 9*x^2],x]

[Out]

((2 + 3*x)*Log[2 + 3*x])/(3*Sqrt[(2 + 3*x)^2])

________________________________________________________________________________________

Maple [A]  time = 0.126, size = 23, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2+3\,x \right ) \ln \left ( 2+3\,x \right ) }{3}{\frac{1}{\sqrt{ \left ( 2+3\,x \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(9*x^2+12*x+4)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.66952, size = 8, normalized size = 0.28 \begin{align*} \frac{1}{3} \, \log \left (x + \frac{2}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+12*x+4)^(1/2),x, algorithm="maxima")

[Out]

1/3*log(x + 2/3)

________________________________________________________________________________________

Fricas [A]  time = 2.18526, size = 24, normalized size = 0.83 \begin{align*} \frac{1}{3} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+12*x+4)^(1/2),x, algorithm="fricas")

[Out]

1/3*log(3*x + 2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{9 x^{2} + 12 x + 4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x**2+12*x+4)**(1/2),x)

[Out]

Integral(1/sqrt(9*x**2 + 12*x + 4), x)

________________________________________________________________________________________

Giac [A]  time = 1.22704, size = 34, normalized size = 1.17 \begin{align*} \frac{\log \left ({\left | 3 \, x + 2 \right |}{\left | \mathrm{sgn}\left (3 \, x + 2\right ) \right |}\right )}{3 \, \mathrm{sgn}\left (3 \, x + 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+12*x+4)^(1/2),x, algorithm="giac")

[Out]

1/3*log(abs(3*x + 2)*abs(sgn(3*x + 2)))/sgn(3*x + 2)