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

Optimal. Leaf size=39 \[ \frac{1}{32 (3 x+2)}+\frac{1}{32 (3 x+2)^2}+\frac{\log (x)}{64}-\frac{1}{64} \log (3 x+2) \]

[Out]

1/(32*(2 + 3*x)^2) + 1/(32*(2 + 3*x)) + Log[x]/64 - Log[2 + 3*x]/64

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Rubi [A]  time = 0.0281361, antiderivative size = 39, 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 \[ \frac{1}{32 (3 x+2)}+\frac{1}{32 (3 x+2)^2}+\frac{\log (x)}{64}-\frac{1}{64} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(4 + 6*x)^3),x]

[Out]

1/(32*(2 + 3*x)^2) + 1/(32*(2 + 3*x)) + Log[x]/64 - Log[2 + 3*x]/64

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Rubi in Sympy [A]  time = 5.34267, size = 29, normalized size = 0.74 \[ \frac{\log{\left (x \right )}}{64} - \frac{\log{\left (3 x + 2 \right )}}{64} + \frac{1}{32 \left (3 x + 2\right )} + \frac{1}{32 \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)/64 - log(3*x + 2)/64 + 1/(32*(3*x + 2)) + 1/(32*(3*x + 2)**2)

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Mathematica [A]  time = 0.0307152, size = 29, normalized size = 0.74 \[ \frac{1}{64} \left (\frac{6 (x+1)}{(3 x+2)^2}+\log (-6 x)-\log (6 x+4)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(4 + 6*x)^3),x]

[Out]

((6*(1 + x))/(2 + 3*x)^2 + Log[-6*x] - Log[4 + 6*x])/64

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Maple [A]  time = 0.01, size = 32, normalized size = 0.8 \[{\frac{1}{32\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{1}{64+96\,x}}+{\frac{\ln \left ( x \right ) }{64}}-{\frac{\ln \left ( 2+3\,x \right ) }{64}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(4+6*x)^3,x)

[Out]

1/32/(2+3*x)^2+1/32/(2+3*x)+1/64*ln(x)-1/64*ln(2+3*x)

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Maxima [A]  time = 1.32844, size = 41, normalized size = 1.05 \[ \frac{3 \,{\left (x + 1\right )}}{32 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{1}{64} \, \log \left (3 \, x + 2\right ) + \frac{1}{64} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/8/((3*x + 2)^3*x),x, algorithm="maxima")

[Out]

3/32*(x + 1)/(9*x^2 + 12*x + 4) - 1/64*log(3*x + 2) + 1/64*log(x)

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Fricas [A]  time = 0.212966, size = 68, normalized size = 1.74 \[ -\frac{{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) -{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (x\right ) - 6 \, x - 6}{64 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/8/((3*x + 2)^3*x),x, algorithm="fricas")

[Out]

-1/64*((9*x^2 + 12*x + 4)*log(3*x + 2) - (9*x^2 + 12*x + 4)*log(x) - 6*x - 6)/(9
*x^2 + 12*x + 4)

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Sympy [A]  time = 0.291737, size = 27, normalized size = 0.69 \[ \frac{3 x + 3}{288 x^{2} + 384 x + 128} + \frac{\log{\left (x \right )}}{64} - \frac{\log{\left (x + \frac{2}{3} \right )}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(4+6*x)**3,x)

[Out]

(3*x + 3)/(288*x**2 + 384*x + 128) + log(x)/64 - log(x + 2/3)/64

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GIAC/XCAS [A]  time = 0.204996, size = 36, normalized size = 0.92 \[ \frac{3 \,{\left (x + 1\right )}}{32 \,{\left (3 \, x + 2\right )}^{2}} - \frac{1}{64} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{1}{64} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/8/((3*x + 2)^3*x),x, algorithm="giac")

[Out]

3/32*(x + 1)/(3*x + 2)^2 - 1/64*ln(abs(3*x + 2)) + 1/64*ln(abs(x))