3.258 \(\int \frac{1}{x^3 (4+6 x)} \, dx\)

Optimal. Leaf size=31 \[ -\frac{1}{8 x^2}+\frac{3}{8 x}+\frac{9 \log (x)}{16}-\frac{9}{16} \log (3 x+2) \]

[Out]

-1/(8*x^2) + 3/(8*x) + (9*Log[x])/16 - (9*Log[2 + 3*x])/16

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Rubi [A]  time = 0.0238058, antiderivative size = 31, 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 \[ -\frac{1}{8 x^2}+\frac{3}{8 x}+\frac{9 \log (x)}{16}-\frac{9}{16} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(4 + 6*x)),x]

[Out]

-1/(8*x^2) + 3/(8*x) + (9*Log[x])/16 - (9*Log[2 + 3*x])/16

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Rubi in Sympy [A]  time = 4.08248, size = 27, normalized size = 0.87 \[ \frac{9 \log{\left (x \right )}}{16} - \frac{9 \log{\left (3 x + 2 \right )}}{16} + \frac{3}{8 x} - \frac{1}{8 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(4+6*x),x)

[Out]

9*log(x)/16 - 9*log(3*x + 2)/16 + 3/(8*x) - 1/(8*x**2)

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Mathematica [A]  time = 0.00393771, size = 31, normalized size = 1. \[ -\frac{1}{8 x^2}+\frac{3}{8 x}+\frac{9 \log (x)}{16}-\frac{9}{16} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(4 + 6*x)),x]

[Out]

-1/(8*x^2) + 3/(8*x) + (9*Log[x])/16 - (9*Log[2 + 3*x])/16

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Maple [A]  time = 0.011, size = 24, normalized size = 0.8 \[ -{\frac{1}{8\,{x}^{2}}}+{\frac{3}{8\,x}}+{\frac{9\,\ln \left ( x \right ) }{16}}-{\frac{9\,\ln \left ( 2+3\,x \right ) }{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(4+6*x),x)

[Out]

-1/8/x^2+3/8/x+9/16*ln(x)-9/16*ln(2+3*x)

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Maxima [A]  time = 1.33352, size = 31, normalized size = 1. \[ \frac{3 \, x - 1}{8 \, x^{2}} - \frac{9}{16} \, \log \left (3 \, x + 2\right ) + \frac{9}{16} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(3*x - 1)/x^2 - 9/16*log(3*x + 2) + 9/16*log(x)

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Fricas [A]  time = 0.206109, size = 38, normalized size = 1.23 \[ -\frac{9 \, x^{2} \log \left (3 \, x + 2\right ) - 9 \, x^{2} \log \left (x\right ) - 6 \, x + 2}{16 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(9*x^2*log(3*x + 2) - 9*x^2*log(x) - 6*x + 2)/x^2

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Sympy [A]  time = 0.243409, size = 26, normalized size = 0.84 \[ \frac{9 \log{\left (x \right )}}{16} - \frac{9 \log{\left (x + \frac{2}{3} \right )}}{16} + \frac{3 x - 1}{8 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(4+6*x),x)

[Out]

9*log(x)/16 - 9*log(x + 2/3)/16 + (3*x - 1)/(8*x**2)

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GIAC/XCAS [A]  time = 0.204247, size = 34, normalized size = 1.1 \[ \frac{3 \, x - 1}{8 \, x^{2}} - \frac{9}{16} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{9}{16} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(3*x - 1)/x^2 - 9/16*ln(abs(3*x + 2)) + 9/16*ln(abs(x))