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

Optimal. Leaf size=60 \[ -\frac {1}{192 x^3}+\frac {9}{256 x^2}-\frac {27}{128 x}-\frac {27}{64 (3 x+2)}-\frac {27}{256 (3 x+2)^2}-\frac {135 \log (x)}{256}+\frac {135}{256} \log (3 x+2) \]

[Out]

-1/192/x^3+9/256/x^2-27/128/x-27/256/(2+3*x)^2-27/64/(2+3*x)-135/256*ln(x)+135/256*ln(2+3*x)

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Rubi [A]  time = 0.02, antiderivative size = 60, 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} \[ \frac {9}{256 x^2}-\frac {1}{192 x^3}-\frac {27}{128 x}-\frac {27}{64 (3 x+2)}-\frac {27}{256 (3 x+2)^2}-\frac {135 \log (x)}{256}+\frac {135}{256} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(4 + 6*x)^3),x]

[Out]

-1/(192*x^3) + 9/(256*x^2) - 27/(128*x) - 27/(256*(2 + 3*x)^2) - 27/(64*(2 + 3*x)) - (135*Log[x])/256 + (135*L
og[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^4 (4+6 x)^3} \, dx &=\int \left (\frac {1}{64 x^4}-\frac {9}{128 x^3}+\frac {27}{128 x^2}-\frac {135}{256 x}+\frac {81}{128 (2+3 x)^3}+\frac {81}{64 (2+3 x)^2}+\frac {405}{256 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{192 x^3}+\frac {9}{256 x^2}-\frac {27}{128 x}-\frac {27}{256 (2+3 x)^2}-\frac {27}{64 (2+3 x)}-\frac {135 \log (x)}{256}+\frac {135}{256} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 49, normalized size = 0.82 \[ \frac {1}{768} \left (-\frac {2 \left (1215 x^4+1215 x^3+180 x^2-30 x+8\right )}{x^3 (3 x+2)^2}-405 \log (x)+405 \log (3 x+2)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(4 + 6*x)^3),x]

[Out]

((-2*(8 - 30*x + 180*x^2 + 1215*x^3 + 1215*x^4))/(x^3*(2 + 3*x)^2) - 405*Log[x] + 405*Log[2 + 3*x])/768

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fricas [A]  time = 0.45, size = 84, normalized size = 1.40 \[ -\frac {2430 \, x^{4} + 2430 \, x^{3} + 360 \, x^{2} - 405 \, {\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )} \log \left (3 \, x + 2\right ) + 405 \, {\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )} \log \relax (x) - 60 \, x + 16}{768 \, {\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(4+6*x)^3,x, algorithm="fricas")

[Out]

-1/768*(2430*x^4 + 2430*x^3 + 360*x^2 - 405*(9*x^5 + 12*x^4 + 4*x^3)*log(3*x + 2) + 405*(9*x^5 + 12*x^4 + 4*x^
3)*log(x) - 60*x + 16)/(9*x^5 + 12*x^4 + 4*x^3)

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giac [A]  time = 1.29, size = 47, normalized size = 0.78 \[ -\frac {1215 \, x^{4} + 1215 \, x^{3} + 180 \, x^{2} - 30 \, x + 8}{384 \, {\left (3 \, x + 2\right )}^{2} x^{3}} + \frac {135}{256} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {135}{256} \, \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(4+6*x)^3,x, algorithm="giac")

[Out]

-1/384*(1215*x^4 + 1215*x^3 + 180*x^2 - 30*x + 8)/((3*x + 2)^2*x^3) + 135/256*log(abs(3*x + 2)) - 135/256*log(
abs(x))

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maple [A]  time = 0.01, size = 47, normalized size = 0.78 \[ -\frac {135 \ln \relax (x )}{256}+\frac {135 \ln \left (3 x +2\right )}{256}-\frac {27}{128 x}+\frac {9}{256 x^{2}}-\frac {1}{192 x^{3}}-\frac {27}{256 \left (3 x +2\right )^{2}}-\frac {27}{64 \left (3 x +2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(4+6*x)^3,x)

[Out]

-1/192/x^3+9/256/x^2-27/128/x-27/256/(3*x+2)^2-27/64/(3*x+2)-135/256*ln(x)+135/256*ln(3*x+2)

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maxima [A]  time = 1.38, size = 53, normalized size = 0.88 \[ -\frac {1215 \, x^{4} + 1215 \, x^{3} + 180 \, x^{2} - 30 \, x + 8}{384 \, {\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )}} + \frac {135}{256} \, \log \left (3 \, x + 2\right ) - \frac {135}{256} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(4+6*x)^3,x, algorithm="maxima")

[Out]

-1/384*(1215*x^4 + 1215*x^3 + 180*x^2 - 30*x + 8)/(9*x^5 + 12*x^4 + 4*x^3) + 135/256*log(3*x + 2) - 135/256*lo
g(x)

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mupad [B]  time = 0.05, size = 47, normalized size = 0.78 \[ \frac {135\,\mathrm {atanh}\left (3\,x+1\right )}{128}-\frac {\frac {45\,x^4}{128}+\frac {45\,x^3}{128}+\frac {5\,x^2}{96}-\frac {5\,x}{576}+\frac {1}{432}}{x^5+\frac {4\,x^4}{3}+\frac {4\,x^3}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(6*x + 4)^3),x)

[Out]

(135*atanh(3*x + 1))/128 - ((5*x^2)/96 - (5*x)/576 + (45*x^3)/128 + (45*x^4)/128 + 1/432)/((4*x^3)/9 + (4*x^4)
/3 + x^5)

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sympy [A]  time = 0.21, size = 51, normalized size = 0.85 \[ - \frac {135 \log {\relax (x )}}{256} + \frac {135 \log {\left (x + \frac {2}{3} \right )}}{256} + \frac {- 1215 x^{4} - 1215 x^{3} - 180 x^{2} + 30 x - 8}{3456 x^{5} + 4608 x^{4} + 1536 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(4+6*x)**3,x)

[Out]

-135*log(x)/256 + 135*log(x + 2/3)/256 + (-1215*x**4 - 1215*x**3 - 180*x**2 + 30*x - 8)/(3456*x**5 + 4608*x**4
+ 1536*x**3)

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