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

Optimal. Leaf size=53 \[ -\frac{1}{128 x^2}+\frac{9}{128 x}+\frac{27}{128 (3 x+2)}+\frac{9}{128 (3 x+2)^2}+\frac{27 \log (x)}{128}-\frac{27}{128} \log (3 x+2) \]

[Out]

-1/(128*x^2) + 9/(128*x) + 9/(128*(2 + 3*x)^2) + 27/(128*(2 + 3*x)) + (27*Log[x]
)/128 - (27*Log[2 + 3*x])/128

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Rubi [A]  time = 0.0387912, antiderivative size = 53, 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 \[ -\frac{1}{128 x^2}+\frac{9}{128 x}+\frac{27}{128 (3 x+2)}+\frac{9}{128 (3 x+2)^2}+\frac{27 \log (x)}{128}-\frac{27}{128} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(4 + 6*x)^3),x]

[Out]

-1/(128*x^2) + 9/(128*x) + 9/(128*(2 + 3*x)^2) + 27/(128*(2 + 3*x)) + (27*Log[x]
)/128 - (27*Log[2 + 3*x])/128

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Rubi in Sympy [A]  time = 6.18605, size = 44, normalized size = 0.83 \[ \frac{27 \log{\left (x \right )}}{128} - \frac{27 \log{\left (3 x + 2 \right )}}{128} + \frac{27}{128 \left (3 x + 2\right )} + \frac{9}{128 \left (3 x + 2\right )^{2}} + \frac{9}{128 x} - \frac{1}{128 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27*log(x)/128 - 27*log(3*x + 2)/128 + 27/(128*(3*x + 2)) + 9/(128*(3*x + 2)**2)
+ 9/(128*x) - 1/(128*x**2)

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Mathematica [A]  time = 0.0438466, size = 44, normalized size = 0.83 \[ \frac{1}{128} \left (\frac{2 \left (81 x^3+81 x^2+12 x-2\right )}{x^2 (3 x+2)^2}+27 \log (x)-27 \log (3 x+2)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*(-2 + 12*x + 81*x^2 + 81*x^3))/(x^2*(2 + 3*x)^2) + 27*Log[x] - 27*Log[2 + 3*
x])/128

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Maple [A]  time = 0.013, size = 42, normalized size = 0.8 \[ -{\frac{1}{128\,{x}^{2}}}+{\frac{9}{128\,x}}+{\frac{9}{128\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{27}{256+384\,x}}+{\frac{27\,\ln \left ( x \right ) }{128}}-{\frac{27\,\ln \left ( 2+3\,x \right ) }{128}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(4+6*x)^3,x)

[Out]

-1/128/x^2+9/128/x+9/128/(2+3*x)^2+27/128/(2+3*x)+27/128*ln(x)-27/128*ln(2+3*x)

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Maxima [A]  time = 1.31641, size = 65, normalized size = 1.23 \[ \frac{81 \, x^{3} + 81 \, x^{2} + 12 \, x - 2}{64 \,{\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )}} - \frac{27}{128} \, \log \left (3 \, x + 2\right ) + \frac{27}{128} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*(81*x^3 + 81*x^2 + 12*x - 2)/(9*x^4 + 12*x^3 + 4*x^2) - 27/128*log(3*x + 2)
+ 27/128*log(x)

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Fricas [A]  time = 0.209024, size = 107, normalized size = 2.02 \[ \frac{162 \, x^{3} + 162 \, x^{2} - 27 \,{\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )} \log \left (3 \, x + 2\right ) + 27 \,{\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )} \log \left (x\right ) + 24 \, x - 4}{128 \,{\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/128*(162*x^3 + 162*x^2 - 27*(9*x^4 + 12*x^3 + 4*x^2)*log(3*x + 2) + 27*(9*x^4
+ 12*x^3 + 4*x^2)*log(x) + 24*x - 4)/(9*x^4 + 12*x^3 + 4*x^2)

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Sympy [A]  time = 0.380459, size = 46, normalized size = 0.87 \[ \frac{27 \log{\left (x \right )}}{128} - \frac{27 \log{\left (x + \frac{2}{3} \right )}}{128} + \frac{81 x^{3} + 81 x^{2} + 12 x - 2}{576 x^{4} + 768 x^{3} + 256 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(4+6*x)**3,x)

[Out]

27*log(x)/128 - 27*log(x + 2/3)/128 + (81*x**3 + 81*x**2 + 12*x - 2)/(576*x**4 +
768*x**3 + 256*x**2)

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GIAC/XCAS [A]  time = 0.204958, size = 58, normalized size = 1.09 \[ \frac{81 \, x^{3} + 81 \, x^{2} + 12 \, x - 2}{64 \,{\left (3 \, x^{2} + 2 \, x\right )}^{2}} - \frac{27}{128} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{27}{128} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*(81*x^3 + 81*x^2 + 12*x - 2)/(3*x^2 + 2*x)^2 - 27/128*ln(abs(3*x + 2)) + 27
/128*ln(abs(x))