3.255 \(\int \frac{1}{x (2+3 x)} \, dx\)

Optimal. Leaf size=17 \[ \frac{\log (x)}{2}-\frac{1}{2} \log (3 x+2) \]

[Out]

Log[x]/2 - Log[2 + 3*x]/2

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Rubi [A]  time = 0.0105639, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\log (x)}{2}-\frac{1}{2} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(2 + 3*x)),x]

[Out]

Log[x]/2 - Log[2 + 3*x]/2

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Rubi in Sympy [A]  time = 2.22471, size = 12, normalized size = 0.71 \[ \frac{\log{\left (x \right )}}{2} - \frac{\log{\left (3 x + 2 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(2+3*x),x)

[Out]

log(x)/2 - log(3*x + 2)/2

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Mathematica [A]  time = 0.00360877, size = 17, normalized size = 1. \[ \frac{\log (x)}{2}-\frac{1}{2} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(2 + 3*x)),x]

[Out]

Log[x]/2 - Log[2 + 3*x]/2

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Maple [A]  time = 0.009, size = 14, normalized size = 0.8 \[{\frac{\ln \left ( x \right ) }{2}}-{\frac{\ln \left ( 2+3\,x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(2+3*x),x)

[Out]

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

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Maxima [A]  time = 1.34183, size = 18, normalized size = 1.06 \[ -\frac{1}{2} \, \log \left (3 \, x + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*log(3*x + 2) + 1/2*log(x)

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Fricas [A]  time = 0.214597, size = 18, normalized size = 1.06 \[ -\frac{1}{2} \, \log \left (3 \, x + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*log(3*x + 2) + 1/2*log(x)

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Sympy [A]  time = 0.16962, size = 12, normalized size = 0.71 \[ \frac{\log{\left (x \right )}}{2} - \frac{\log{\left (x + \frac{2}{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(2+3*x),x)

[Out]

log(x)/2 - log(x + 2/3)/2

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GIAC/XCAS [A]  time = 0.202598, size = 20, normalized size = 1.18 \[ -\frac{1}{2} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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