3.257 \(\int \frac{1}{x^2 (4+6 x)} \, dx\)

Optimal. Leaf size=24 \[ -\frac{1}{4 x}-\frac{3 \log (x)}{8}+\frac{3}{8} \log (3 x+2) \]

[Out]

-1/(4*x) - (3*Log[x])/8 + (3*Log[2 + 3*x])/8

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Rubi [A]  time = 0.0216753, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{1}{4 x}-\frac{3 \log (x)}{8}+\frac{3}{8} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(4 + 6*x)),x]

[Out]

-1/(4*x) - (3*Log[x])/8 + (3*Log[2 + 3*x])/8

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Rubi in Sympy [A]  time = 3.58562, size = 20, normalized size = 0.83 \[ - \frac{3 \log{\left (x \right )}}{8} + \frac{3 \log{\left (3 x + 2 \right )}}{8} - \frac{1}{4 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(4+6*x),x)

[Out]

-3*log(x)/8 + 3*log(3*x + 2)/8 - 1/(4*x)

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Mathematica [A]  time = 0.00391467, size = 24, normalized size = 1. \[ -\frac{1}{4 x}-\frac{3 \log (x)}{8}+\frac{3}{8} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(4 + 6*x)),x]

[Out]

-1/(4*x) - (3*Log[x])/8 + (3*Log[2 + 3*x])/8

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Maple [A]  time = 0.01, size = 19, normalized size = 0.8 \[ -{\frac{1}{4\,x}}-{\frac{3\,\ln \left ( x \right ) }{8}}+{\frac{3\,\ln \left ( 2+3\,x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(4+6*x),x)

[Out]

-1/4/x-3/8*ln(x)+3/8*ln(2+3*x)

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Maxima [A]  time = 1.34301, size = 24, normalized size = 1. \[ -\frac{1}{4 \, x} + \frac{3}{8} \, \log \left (3 \, x + 2\right ) - \frac{3}{8} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/x + 3/8*log(3*x + 2) - 3/8*log(x)

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Fricas [A]  time = 0.211774, size = 28, normalized size = 1.17 \[ \frac{3 \, x \log \left (3 \, x + 2\right ) - 3 \, x \log \left (x\right ) - 2}{8 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(3*x*log(3*x + 2) - 3*x*log(x) - 2)/x

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Sympy [A]  time = 0.210732, size = 20, normalized size = 0.83 \[ - \frac{3 \log{\left (x \right )}}{8} + \frac{3 \log{\left (x + \frac{2}{3} \right )}}{8} - \frac{1}{4 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(4+6*x),x)

[Out]

-3*log(x)/8 + 3*log(x + 2/3)/8 - 1/(4*x)

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GIAC/XCAS [A]  time = 0.207361, size = 27, normalized size = 1.12 \[ -\frac{1}{4 \, x} + \frac{3}{8} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - \frac{3}{8} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/x + 3/8*ln(abs(3*x + 2)) - 3/8*ln(abs(x))