3.260 \(\int \frac{1}{x^5 (4+6 x)} \, dx\)

Optimal. Leaf size=45 \[ -\frac{1}{16 x^4}+\frac{1}{8 x^3}-\frac{9}{32 x^2}+\frac{27}{32 x}+\frac{81 \log (x)}{64}-\frac{81}{64} \log (3 x+2) \]

[Out]

-1/(16*x^4) + 1/(8*x^3) - 9/(32*x^2) + 27/(32*x) + (81*Log[x])/64 - (81*Log[2 +
3*x])/64

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Rubi [A]  time = 0.03064, antiderivative size = 45, 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 \[ -\frac{1}{16 x^4}+\frac{1}{8 x^3}-\frac{9}{32 x^2}+\frac{27}{32 x}+\frac{81 \log (x)}{64}-\frac{81}{64} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(4 + 6*x)),x]

[Out]

-1/(16*x^4) + 1/(8*x^3) - 9/(32*x^2) + 27/(32*x) + (81*Log[x])/64 - (81*Log[2 +
3*x])/64

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Rubi in Sympy [A]  time = 4.85974, size = 41, normalized size = 0.91 \[ \frac{81 \log{\left (x \right )}}{64} - \frac{81 \log{\left (3 x + 2 \right )}}{64} + \frac{27}{32 x} - \frac{9}{32 x^{2}} + \frac{1}{8 x^{3}} - \frac{1}{16 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(4+6*x),x)

[Out]

81*log(x)/64 - 81*log(3*x + 2)/64 + 27/(32*x) - 9/(32*x**2) + 1/(8*x**3) - 1/(16
*x**4)

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Mathematica [A]  time = 0.00414986, size = 45, normalized size = 1. \[ -\frac{1}{16 x^4}+\frac{1}{8 x^3}-\frac{9}{32 x^2}+\frac{27}{32 x}+\frac{81 \log (x)}{64}-\frac{81}{64} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(4 + 6*x)),x]

[Out]

-1/(16*x^4) + 1/(8*x^3) - 9/(32*x^2) + 27/(32*x) + (81*Log[x])/64 - (81*Log[2 +
3*x])/64

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Maple [A]  time = 0.01, size = 34, normalized size = 0.8 \[ -{\frac{1}{16\,{x}^{4}}}+{\frac{1}{8\,{x}^{3}}}-{\frac{9}{32\,{x}^{2}}}+{\frac{27}{32\,x}}+{\frac{81\,\ln \left ( x \right ) }{64}}-{\frac{81\,\ln \left ( 2+3\,x \right ) }{64}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(4+6*x),x)

[Out]

-1/16/x^4+1/8/x^3-9/32/x^2+27/32/x+81/64*ln(x)-81/64*ln(2+3*x)

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Maxima [A]  time = 1.34691, size = 45, normalized size = 1. \[ \frac{27 \, x^{3} - 9 \, x^{2} + 4 \, x - 2}{32 \, x^{4}} - \frac{81}{64} \, \log \left (3 \, x + 2\right ) + \frac{81}{64} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2/((3*x + 2)*x^5),x, algorithm="maxima")

[Out]

1/32*(27*x^3 - 9*x^2 + 4*x - 2)/x^4 - 81/64*log(3*x + 2) + 81/64*log(x)

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Fricas [A]  time = 0.212699, size = 51, normalized size = 1.13 \[ -\frac{81 \, x^{4} \log \left (3 \, x + 2\right ) - 81 \, x^{4} \log \left (x\right ) - 54 \, x^{3} + 18 \, x^{2} - 8 \, x + 4}{64 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2/((3*x + 2)*x^5),x, algorithm="fricas")

[Out]

-1/64*(81*x^4*log(3*x + 2) - 81*x^4*log(x) - 54*x^3 + 18*x^2 - 8*x + 4)/x^4

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Sympy [A]  time = 0.298061, size = 36, normalized size = 0.8 \[ \frac{81 \log{\left (x \right )}}{64} - \frac{81 \log{\left (x + \frac{2}{3} \right )}}{64} + \frac{27 x^{3} - 9 x^{2} + 4 x - 2}{32 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(4+6*x),x)

[Out]

81*log(x)/64 - 81*log(x + 2/3)/64 + (27*x**3 - 9*x**2 + 4*x - 2)/(32*x**4)

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GIAC/XCAS [A]  time = 0.203864, size = 47, normalized size = 1.04 \[ \frac{27 \, x^{3} - 9 \, x^{2} + 4 \, x - 2}{32 \, x^{4}} - \frac{81}{64} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{81}{64} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2/((3*x + 2)*x^5),x, algorithm="giac")

[Out]

1/32*(27*x^3 - 9*x^2 + 4*x - 2)/x^4 - 81/64*ln(abs(3*x + 2)) + 81/64*ln(abs(x))