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

Optimal. Leaf size=35 \[ -\frac{1}{16 x}-\frac{3}{16 (3 x+2)}-\frac{3 \log (x)}{16}+\frac{3}{16} \log (3 x+2) \]

[Out]

-1/(16*x) - 3/(16*(2 + 3*x)) - (3*Log[x])/16 + (3*Log[2 + 3*x])/16

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Rubi [A]  time = 0.0253209, antiderivative size = 35, 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 \[ -\frac{1}{16 x}-\frac{3}{16 (3 x+2)}-\frac{3 \log (x)}{16}+\frac{3}{16} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(4 + 6*x)^2),x]

[Out]

-1/(16*x) - 3/(16*(2 + 3*x)) - (3*Log[x])/16 + (3*Log[2 + 3*x])/16

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Rubi in Sympy [A]  time = 4.48447, size = 27, normalized size = 0.77 \[ - \frac{3 \log{\left (x \right )}}{16} + \frac{3 \log{\left (3 x + 2 \right )}}{16} - \frac{3}{16 \left (3 x + 2\right )} - \frac{1}{16 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*log(x)/16 + 3*log(3*x + 2)/16 - 3/(16*(3*x + 2)) - 1/(16*x)

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Mathematica [A]  time = 0.0218254, size = 31, normalized size = 0.89 \[ \frac{1}{16} \left (-\frac{1}{x}-\frac{3}{3 x+2}-3 \log (x)+3 \log (3 x+2)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-x^(-1) - 3/(2 + 3*x) - 3*Log[x] + 3*Log[2 + 3*x])/16

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Maple [A]  time = 0.014, size = 28, normalized size = 0.8 \[ -{\frac{1}{16\,x}}-{\frac{3}{32+48\,x}}-{\frac{3\,\ln \left ( x \right ) }{16}}+{\frac{3\,\ln \left ( 2+3\,x \right ) }{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(4+6*x)^2,x)

[Out]

-1/16/x-3/16/(2+3*x)-3/16*ln(x)+3/16*ln(2+3*x)

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Maxima [A]  time = 1.31772, size = 42, normalized size = 1.2 \[ -\frac{3 \, x + 1}{8 \,{\left (3 \, x^{2} + 2 \, x\right )}} + \frac{3}{16} \, \log \left (3 \, x + 2\right ) - \frac{3}{16} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/4/((3*x + 2)^2*x^2),x, algorithm="maxima")

[Out]

-1/8*(3*x + 1)/(3*x^2 + 2*x) + 3/16*log(3*x + 2) - 3/16*log(x)

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Fricas [A]  time = 0.208669, size = 65, normalized size = 1.86 \[ \frac{3 \,{\left (3 \, x^{2} + 2 \, x\right )} \log \left (3 \, x + 2\right ) - 3 \,{\left (3 \, x^{2} + 2 \, x\right )} \log \left (x\right ) - 6 \, x - 2}{16 \,{\left (3 \, x^{2} + 2 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/4/((3*x + 2)^2*x^2),x, algorithm="fricas")

[Out]

1/16*(3*(3*x^2 + 2*x)*log(3*x + 2) - 3*(3*x^2 + 2*x)*log(x) - 6*x - 2)/(3*x^2 +
2*x)

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Sympy [A]  time = 0.264752, size = 29, normalized size = 0.83 \[ - \frac{3 x + 1}{24 x^{2} + 16 x} - \frac{3 \log{\left (x \right )}}{16} + \frac{3 \log{\left (x + \frac{2}{3} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(4+6*x)**2,x)

[Out]

-(3*x + 1)/(24*x**2 + 16*x) - 3*log(x)/16 + 3*log(x + 2/3)/16

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GIAC/XCAS [A]  time = 0.2062, size = 54, normalized size = 1.54 \[ -\frac{3}{16 \,{\left (3 \, x + 2\right )}} + \frac{3}{32 \,{\left (\frac{2}{3 \, x + 2} - 1\right )}} - \frac{3}{16} \,{\rm ln}\left ({\left | -\frac{2}{3 \, x + 2} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/4/((3*x + 2)^2*x^2),x, algorithm="giac")

[Out]

-3/16/(3*x + 2) + 3/32/(2/(3*x + 2) - 1) - 3/16*ln(abs(-2/(3*x + 2) + 1))