3.2408 \(\int \frac{1}{x \sqrt{4+12 x+9 x^2}} \, dx\)

Optimal. Leaf size=27 \[ -\frac{(3 x+2) \tanh ^{-1}(3 x+1)}{\sqrt{9 x^2+12 x+4}} \]

[Out]

-(((2 + 3*x)*ArcTanh[1 + 3*x])/Sqrt[4 + 12*x + 9*x^2])

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Rubi [B]  time = 0.0441586, antiderivative size = 55, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{(3 x+2) \log (x)}{2 \sqrt{9 x^2+12 x+4}}-\frac{(3 x+2) \log (3 x+2)}{2 \sqrt{9 x^2+12 x+4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[4 + 12*x + 9*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.38539, size = 49, normalized size = 1.81 \[ \frac{3 \sqrt{9 x^{2} + 12 x + 4} \log{\left (x \right )}}{2 \left (9 x + 6\right )} - \frac{3 \sqrt{9 x^{2} + 12 x + 4} \log{\left (3 x + 2 \right )}}{2 \left (9 x + 6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/((2+3*x)**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0224727, size = 31, normalized size = 1.15 \[ \frac{(3 x+2) (\log (x)-\log (3 x+2))}{2 \sqrt{(3 x+2)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[4 + 12*x + 9*x^2]),x]

[Out]

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

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Maple [A]  time = 0.013, size = 28, normalized size = 1. \[{\frac{ \left ( 2+3\,x \right ) \left ( \ln \left ( x \right ) -\ln \left ( 2+3\,x \right ) \right ) }{2}{\frac{1}{\sqrt{ \left ( 2+3\,x \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/((2+3*x)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.747181, size = 32, normalized size = 1.19 \[ -\frac{1}{2} \, \left (-1\right )^{12 \, x + 8} \log \left (\frac{12 \, x}{{\left | x \right |}} + \frac{8}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((3*x + 2)^2)*x),x, algorithm="maxima")

[Out]

-1/2*(-1)^(12*x + 8)*log(12*x/abs(x) + 8/abs(x))

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Fricas [A]  time = 0.225351, size = 18, normalized size = 0.67 \[ -\frac{1}{2} \, \log \left (3 \, x + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((3*x + 2)^2)*x),x, algorithm="fricas")

[Out]

-1/2*log(3*x + 2) + 1/2*log(x)

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Sympy [A]  time = 0.227695, size = 12, normalized size = 0.44 \[ \frac{\log{\left (x \right )}}{2} - \frac{\log{\left (x + \frac{2}{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/((2+3*x)**2)**(1/2),x)

[Out]

log(x)/2 - log(x + 2/3)/2

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GIAC/XCAS [A]  time = 0.206516, size = 28, normalized size = 1.04 \[ -\frac{1}{2} \,{\left ({\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) -{\rm ln}\left ({\left | x \right |}\right )\right )}{\rm sign}\left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((3*x + 2)^2)*x),x, algorithm="giac")

[Out]

-1/2*(ln(abs(3*x + 2)) - ln(abs(x)))*sign(3*x + 2)