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

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

[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*L
og[x])/1024 - (1215*Log[2 + 3*x])/1024

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Rubi [A]  time = 0.0223792, 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, Rules used = {44} \[ -\frac{27}{256 x^2}+\frac{3}{128 x^3}-\frac{1}{256 x^4}+\frac{135}{256 x}+\frac{405}{512 (3 x+2)}+\frac{81}{512 (3 x+2)^2}+\frac{1215 \log (x)}{1024}-\frac{1215 \log (3 x+2)}{1024} \]

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*L
og[x])/1024 - (1215*Log[2 + 3*x])/1024

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)^3} \, dx &=\int \left (\frac{1}{64 x^5}-\frac{9}{128 x^4}+\frac{27}{128 x^3}-\frac{135}{256 x^2}+\frac{1215}{1024 x}-\frac{243}{256 (2+3 x)^3}-\frac{1215}{512 (2+3 x)^2}-\frac{3645}{1024 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{256 x^4}+\frac{3}{128 x^3}-\frac{27}{256 x^2}+\frac{135}{256 x}+\frac{81}{512 (2+3 x)^2}+\frac{405}{512 (2+3 x)}+\frac{1215 \log (x)}{1024}-\frac{1215 \log (2+3 x)}{1024}\\ \end{align*}

Mathematica [A]  time = 0.032343, 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) + 1215*Log[x] - 1215*Log[2 + 3*x])
/1024

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Maple [A]  time = 0.01, size = 52, normalized size = 0.8 \begin{align*} -{\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}} \end{align*}

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 = 1.07264, size = 78, normalized size = 1.16 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(4+6*x)^3,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 = 1.55242, size = 232, normalized size = 3.46 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(4+6*x)^3,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 = 0.18851, size = 56, normalized size = 0.84 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.18215, size = 70, normalized size = 1.04 \begin{align*} \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} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{1215}{1024} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(4+6*x)^3,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) - 1215/1024*log(abs(3*x + 2)) + 12
15/1024*log(abs(x))