3.3.65 \(\int \frac {1}{x^5 (4+6 x)^2} \, dx\) [265]

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

[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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Rubi [A]
time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 = {46} \begin {gather*} -\frac {1}{64 x^4}+\frac {1}{16 x^3}-\frac {27}{128 x^2}+\frac {27}{32 x}+\frac {81}{128 (3 x+2)}+\frac {405 \log (x)}{256}-\frac {405}{256} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(4 + 6*x)^2),x]

[Out]

-1/64*1/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 46

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] && Lt
Q[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*}

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Mathematica [A]
time = 0.01, size = 56, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(4 + 6*x)^2),x]

[Out]

-1/64*1/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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Mathics [A]
time = 1.91, size = 51, normalized size = 0.91 \begin {gather*} \frac {-8+20 x-60 x^2+270 x^3+405 x^4 \left (2+3 x\right ) \left (\text {Log}\left [x\right ]-\text {Log}\left [\frac {2}{3}+x\right ]\right )+810 x^4}{256 x^4 \left (2+3 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(x^5*(4 + 6*x)^2),x]')

[Out]

(-8 + 20 x - 60 x ^ 2 + 270 x ^ 3 + 405 x ^ 4 (2 + 3 x) (Log[x] - Log[2 / 3 + x]) + 810 x ^ 4) / (256 x ^ 4 (2
+ 3 x))

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Maple [A]
time = 0.11, size = 43, normalized size = 0.77

method	result	size
default	\(-\frac {1}{64 x^{4}}+\frac {1}{16 x^{3}}-\frac {27}{128 x^{2}}+\frac {27}{32 x}+\frac {81}{128 \left (2+3 x \right )}+\frac {405 \ln \left (x \right )}{256}-\frac {405 \ln \left (2+3 x \right )}{256}\)	\(43\)
norman	\(\frac {-\frac {1}{32}-\frac {1215}{256} x^{5}+\frac {5}{64} x -\frac {15}{64} x^{2}+\frac {135}{128} x^{3}}{x^{4} \left (2+3 x \right )}+\frac {405 \ln \left (x \right )}{256}-\frac {405 \ln \left (2+3 x \right )}{256}\)	\(45\)
risch	\(\frac {\frac {405}{128} x^{4}+\frac {135}{128} x^{3}-\frac {15}{64} x^{2}+\frac {5}{64} x -\frac {1}{32}}{x^{4} \left (2+3 x \right )}+\frac {405 \ln \left (x \right )}{256}-\frac {405 \ln \left (2+3 x \right )}{256}\)	\(46\)
meijerg	\(-\frac {1}{64 x^{4}}+\frac {1}{16 x^{3}}-\frac {27}{128 x^{2}}+\frac {27}{32 x}+\frac {81}{256}+\frac {405 \ln \left (x \right )}{256}-\frac {405 \ln \left (2\right )}{256}+\frac {405 \ln \left (3\right )}{256}-\frac {729 x}{256 \left (9 x +6\right )}-\frac {405 \ln \left (1+\frac {3 x}{2}\right )}{256}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.24, size = 48, normalized size = 0.86 \begin {gather*} \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 {gather*}

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 = 0.31, size = 69, normalized size = 1.23 \begin {gather*} \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 {gather*}

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 = 0.08, size = 46, normalized size = 0.82 \begin {gather*} \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 {gather*}

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 = 0.00, size = 54, normalized size = 0.96 \begin {gather*} -\frac {405}{256} \ln \left |3 x+2\right |+\frac {405}{256} \ln \left |x\right |+\frac {\frac {1}{1024} \left (3240 x^{4}+1080 x^{3}-240 x^{2}+80 x-32\right )}{x^{4} \left (3 x+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.09, size = 41, normalized size = 0.73 \begin {gather*} \frac {\frac {135\,x^4}{128}+\frac {45\,x^3}{128}-\frac {5\,x^2}{64}+\frac {5\,x}{192}-\frac {1}{96}}{x^5+\frac {2\,x^4}{3}}-\frac {405\,\mathrm {atanh}\left (3\,x+1\right )}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(6*x + 4)^2),x)

[Out]

((5*x)/192 - (5*x^2)/64 + (45*x^3)/128 + (135*x^4)/128 - 1/96)/((2*x^4)/3 + x^5) - (405*atanh(3*x + 1))/128

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