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

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

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Rubi [A]  time = 0.02, antiderivative size = 67, 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 = {44} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 54, normalized size = 0.81 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^5 (4+6 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.99, size = 89, normalized size = 1.33 \begin {gather*} \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 \relax (x) + 48 \, x - 16}{1024 \, {\left (9 \, x^{6} + 12 \, x^{5} + 4 \, x^{4}\right )}} \end {gather*}

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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giac [A]  time = 1.13, size = 52, normalized size = 0.78 \begin {gather*} \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 {gather*}

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))

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maple [A]  time = 0.01, size = 52, normalized size = 0.78 \begin {gather*} \frac {1215 \ln \relax (x )}{1024}-\frac {1215 \ln \left (3 x +2\right )}{1024}+\frac {135}{256 x}-\frac {27}{256 x^{2}}+\frac {3}{128 x^{3}}-\frac {1}{256 x^{4}}+\frac {81}{512 \left (3 x +2\right )^{2}}+\frac {405}{512 \left (3 x +2\right )} \end {gather*}

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/(3*x+2)^2+405/512/(3*x+2)+1215/1024*ln(x)-1215/1024*ln(3*x+2)

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maxima [A]  time = 1.39, size = 58, normalized size = 0.87 \begin {gather*} \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 \relax (x) \end {gather*}

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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mupad [B]  time = 0.05, size = 51, normalized size = 0.76 \begin {gather*} \frac {\frac {405\,x^5}{512}+\frac {405\,x^4}{512}+\frac {15\,x^3}{128}-\frac {5\,x^2}{256}+\frac {x}{192}-\frac {1}{576}}{x^6+\frac {4\,x^5}{3}+\frac {4\,x^4}{9}}-\frac {1215\,\mathrm {atanh}\left (3\,x+1\right )}{512} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(6*x + 4)^3),x)

[Out]

(x/192 - (5*x^2)/256 + (15*x^3)/128 + (405*x^4)/512 + (405*x^5)/512 - 1/576)/((4*x^4)/9 + (4*x^5)/3 + x^6) - (
1215*atanh(3*x + 1))/512

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sympy [A]  time = 0.22, size = 56, normalized size = 0.84 \begin {gather*} \frac {1215 \log {\relax (x )}}{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 {gather*}

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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