3.265 \(\int \frac{1}{x^5 (4+6 x)^2} \, dx\)

Optimal. Leaf size=56 \[ -\frac{27}{128 x^2}+\frac{1}{16 x^3}-\frac{1}{64 x^4}+\frac{27}{32 x}+\frac{81}{128 (3 x+2)}+\frac{405 \log (x)}{256}-\frac{405}{256} \log (3 x+2) \]

[Out]

-1/(64*x^4) + 1/(16*x^3) - 27/(128*x^2) + 27/(32*x) + 81/(128*(2 + 3*x)) + (405*Log[x])/256 - (405*Log[2 + 3*x
])/256

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Rubi [A]  time = 0.019642, antiderivative size = 56, 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, Rules used = {44} \[ -\frac{27}{128 x^2}+\frac{1}{16 x^3}-\frac{1}{64 x^4}+\frac{27}{32 x}+\frac{81}{128 (3 x+2)}+\frac{405 \log (x)}{256}-\frac{405}{256} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(4 + 6*x)^2),x]

[Out]

-1/(64*x^4) + 1/(16*x^3) - 27/(128*x^2) + 27/(32*x) + 81/(128*(2 + 3*x)) + (405*Log[x])/256 - (405*Log[2 + 3*x
])/256

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^5 (4+6 x)^2} \, dx &=\int \left (\frac{1}{16 x^5}-\frac{3}{16 x^4}+\frac{27}{64 x^3}-\frac{27}{32 x^2}+\frac{405}{256 x}-\frac{243}{128 (2+3 x)^2}-\frac{1215}{256 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{64 x^4}+\frac{1}{16 x^3}-\frac{27}{128 x^2}+\frac{27}{32 x}+\frac{81}{128 (2+3 x)}+\frac{405 \log (x)}{256}-\frac{405}{256} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0178101, size = 56, normalized size = 1. \[ -\frac{27}{128 x^2}+\frac{1}{16 x^3}-\frac{1}{64 x^4}+\frac{27}{32 x}+\frac{81}{128 (3 x+2)}+\frac{405 \log (x)}{256}-\frac{405}{256} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(4 + 6*x)^2),x]

[Out]

-1/(64*x^4) + 1/(16*x^3) - 27/(128*x^2) + 27/(32*x) + 81/(128*(2 + 3*x)) + (405*Log[x])/256 - (405*Log[2 + 3*x
])/256

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Maple [A]  time = 0.01, size = 43, normalized size = 0.8 \begin{align*} -{\frac{1}{64\,{x}^{4}}}+{\frac{1}{16\,{x}^{3}}}-{\frac{27}{128\,{x}^{2}}}+{\frac{27}{32\,x}}+{\frac{81}{256+384\,x}}+{\frac{405\,\ln \left ( x \right ) }{256}}-{\frac{405\,\ln \left ( 2+3\,x \right ) }{256}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(4+6*x)^2,x)

[Out]

-1/64/x^4+1/16/x^3-27/128/x^2+27/32/x+81/128/(2+3*x)+405/256*ln(x)-405/256*ln(2+3*x)

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Maxima [A]  time = 1.04453, size = 65, normalized size = 1.16 \begin{align*} \frac{405 \, x^{4} + 135 \, x^{3} - 30 \, x^{2} + 10 \, x - 4}{128 \,{\left (3 \, x^{5} + 2 \, x^{4}\right )}} - \frac{405}{256} \, \log \left (3 \, x + 2\right ) + \frac{405}{256} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(4+6*x)^2,x, algorithm="maxima")

[Out]

1/128*(405*x^4 + 135*x^3 - 30*x^2 + 10*x - 4)/(3*x^5 + 2*x^4) - 405/256*log(3*x + 2) + 405/256*log(x)

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Fricas [A]  time = 1.83577, size = 171, normalized size = 3.05 \begin{align*} \frac{810 \, x^{4} + 270 \, x^{3} - 60 \, x^{2} - 405 \,{\left (3 \, x^{5} + 2 \, x^{4}\right )} \log \left (3 \, x + 2\right ) + 405 \,{\left (3 \, x^{5} + 2 \, x^{4}\right )} \log \left (x\right ) + 20 \, x - 8}{256 \,{\left (3 \, x^{5} + 2 \, x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(4+6*x)^2,x, algorithm="fricas")

[Out]

1/256*(810*x^4 + 270*x^3 - 60*x^2 - 405*(3*x^5 + 2*x^4)*log(3*x + 2) + 405*(3*x^5 + 2*x^4)*log(x) + 20*x - 8)/
(3*x^5 + 2*x^4)

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Sympy [A]  time = 0.172982, size = 46, normalized size = 0.82 \begin{align*} \frac{405 \log{\left (x \right )}}{256} - \frac{405 \log{\left (x + \frac{2}{3} \right )}}{256} + \frac{405 x^{4} + 135 x^{3} - 30 x^{2} + 10 x - 4}{384 x^{5} + 256 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(4+6*x)**2,x)

[Out]

405*log(x)/256 - 405*log(x + 2/3)/256 + (405*x**4 + 135*x**3 - 30*x**2 + 10*x - 4)/(384*x**5 + 256*x**4)

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Giac [A]  time = 1.16018, size = 93, normalized size = 1.66 \begin{align*} \frac{81}{128 \,{\left (3 \, x + 2\right )}} - \frac{27 \,{\left (\frac{520}{3 \, x + 2} - \frac{1200}{{\left (3 \, x + 2\right )}^{2}} + \frac{960}{{\left (3 \, x + 2\right )}^{3}} - 77\right )}}{1024 \,{\left (\frac{2}{3 \, x + 2} - 1\right )}^{4}} + \frac{405}{256} \, \log \left ({\left | -\frac{2}{3 \, x + 2} + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(4+6*x)^2,x, algorithm="giac")

[Out]

81/128/(3*x + 2) - 27/1024*(520/(3*x + 2) - 1200/(3*x + 2)^2 + 960/(3*x + 2)^3 - 77)/(2/(3*x + 2) - 1)^4 + 405
/256*log(abs(-2/(3*x + 2) + 1))