3.270 \(\int \frac{1}{x^5 (4+6 x)^3} \, dx\)

Optimal. Leaf size=67 \[ -\frac{1}{256 x^4}+\frac{3}{128 x^3}-\frac{27}{256 x^2}+\frac{135}{256 x}+\frac{405}{512 (3 x+2)}+\frac{81}{512 (3 x+2)^2}+\frac{1215 \log (x)}{1024}-\frac{1215 \log (3 x+2)}{1024} \]

[Out]

-1/(256*x^4) + 3/(128*x^3) - 27/(256*x^2) + 135/(256*x) + 81/(512*(2 + 3*x)^2) +
405/(512*(2 + 3*x)) + (1215*Log[x])/1024 - (1215*Log[2 + 3*x])/1024

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Rubi [A]  time = 0.051691, antiderivative size = 67, 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 \[ -\frac{1}{256 x^4}+\frac{3}{128 x^3}-\frac{27}{256 x^2}+\frac{135}{256 x}+\frac{405}{512 (3 x+2)}+\frac{81}{512 (3 x+2)^2}+\frac{1215 \log (x)}{1024}-\frac{1215 \log (3 x+2)}{1024} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(4 + 6*x)^3),x]

[Out]

-1/(256*x^4) + 3/(128*x^3) - 27/(256*x^2) + 135/(256*x) + 81/(512*(2 + 3*x)^2) +
405/(512*(2 + 3*x)) + (1215*Log[x])/1024 - (1215*Log[2 + 3*x])/1024

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Rubi in Sympy [A]  time = 7.46379, size = 58, normalized size = 0.87 \[ \frac{1215 \log{\left (x \right )}}{1024} - \frac{1215 \log{\left (3 x + 2 \right )}}{1024} + \frac{405}{512 \left (3 x + 2\right )} + \frac{81}{512 \left (3 x + 2\right )^{2}} + \frac{135}{256 x} - \frac{27}{256 x^{2}} + \frac{3}{128 x^{3}} - \frac{1}{256 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(4+6*x)**3,x)

[Out]

1215*log(x)/1024 - 1215*log(3*x + 2)/1024 + 405/(512*(3*x + 2)) + 81/(512*(3*x +
2)**2) + 135/(256*x) - 27/(256*x**2) + 3/(128*x**3) - 1/(256*x**4)

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Mathematica [A]  time = 0.0325532, size = 54, normalized size = 0.81 \[ \frac{\frac{2 \left (3645 x^5+3645 x^4+540 x^3-90 x^2+24 x-8\right )}{x^4 (3 x+2)^2}+1215 \log (x)-1215 \log (3 x+2)}{1024} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(4 + 6*x)^3),x]

[Out]

((2*(-8 + 24*x - 90*x^2 + 540*x^3 + 3645*x^4 + 3645*x^5))/(x^4*(2 + 3*x)^2) + 12
15*Log[x] - 1215*Log[2 + 3*x])/1024

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Maple [A]  time = 0.013, size = 52, normalized size = 0.8 \[ -{\frac{1}{256\,{x}^{4}}}+{\frac{3}{128\,{x}^{3}}}-{\frac{27}{256\,{x}^{2}}}+{\frac{135}{256\,x}}+{\frac{81}{512\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{405}{1024+1536\,x}}+{\frac{1215\,\ln \left ( x \right ) }{1024}}-{\frac{1215\,\ln \left ( 2+3\,x \right ) }{1024}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(4+6*x)^3,x)

[Out]

-1/256/x^4+3/128/x^3-27/256/x^2+135/256/x+81/512/(2+3*x)^2+405/512/(2+3*x)+1215/
1024*ln(x)-1215/1024*ln(2+3*x)

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Maxima [A]  time = 1.33411, size = 78, normalized size = 1.16 \[ \frac{3645 \, x^{5} + 3645 \, x^{4} + 540 \, x^{3} - 90 \, x^{2} + 24 \, x - 8}{512 \,{\left (9 \, x^{6} + 12 \, x^{5} + 4 \, x^{4}\right )}} - \frac{1215}{1024} \, \log \left (3 \, x + 2\right ) + \frac{1215}{1024} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/8/((3*x + 2)^3*x^5),x, algorithm="maxima")

[Out]

1/512*(3645*x^5 + 3645*x^4 + 540*x^3 - 90*x^2 + 24*x - 8)/(9*x^6 + 12*x^5 + 4*x^
4) - 1215/1024*log(3*x + 2) + 1215/1024*log(x)

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Fricas [A]  time = 0.20994, size = 120, normalized size = 1.79 \[ \frac{7290 \, x^{5} + 7290 \, x^{4} + 1080 \, x^{3} - 180 \, x^{2} - 1215 \,{\left (9 \, x^{6} + 12 \, x^{5} + 4 \, x^{4}\right )} \log \left (3 \, x + 2\right ) + 1215 \,{\left (9 \, x^{6} + 12 \, x^{5} + 4 \, x^{4}\right )} \log \left (x\right ) + 48 \, x - 16}{1024 \,{\left (9 \, x^{6} + 12 \, x^{5} + 4 \, x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/8/((3*x + 2)^3*x^5),x, algorithm="fricas")

[Out]

1/1024*(7290*x^5 + 7290*x^4 + 1080*x^3 - 180*x^2 - 1215*(9*x^6 + 12*x^5 + 4*x^4)
*log(3*x + 2) + 1215*(9*x^6 + 12*x^5 + 4*x^4)*log(x) + 48*x - 16)/(9*x^6 + 12*x^
5 + 4*x^4)

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Sympy [A]  time = 0.443629, size = 56, normalized size = 0.84 \[ \frac{1215 \log{\left (x \right )}}{1024} - \frac{1215 \log{\left (x + \frac{2}{3} \right )}}{1024} + \frac{3645 x^{5} + 3645 x^{4} + 540 x^{3} - 90 x^{2} + 24 x - 8}{4608 x^{6} + 6144 x^{5} + 2048 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(4+6*x)**3,x)

[Out]

1215*log(x)/1024 - 1215*log(x + 2/3)/1024 + (3645*x**5 + 3645*x**4 + 540*x**3 -
90*x**2 + 24*x - 8)/(4608*x**6 + 6144*x**5 + 2048*x**4)

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GIAC/XCAS [A]  time = 0.20539, size = 70, normalized size = 1.04 \[ \frac{3645 \, x^{5} + 3645 \, x^{4} + 540 \, x^{3} - 90 \, x^{2} + 24 \, x - 8}{512 \,{\left (3 \, x + 2\right )}^{2} x^{4}} - \frac{1215}{1024} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{1215}{1024} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/8/((3*x + 2)^3*x^5),x, algorithm="giac")

[Out]

1/512*(3645*x^5 + 3645*x^4 + 540*x^3 - 90*x^2 + 24*x - 8)/((3*x + 2)^2*x^4) - 12
15/1024*ln(abs(3*x + 2)) + 1215/1024*ln(abs(x))