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

Optimal. Leaf size=53 \[ -\frac {1}{128 x^2}+\frac {9}{128 x}+\frac {27}{128 (3 x+2)}+\frac {9}{128 (3 x+2)^2}+\frac {27 \log (x)}{128}-\frac {27}{128} \log (3 x+2) \]

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Rubi [A]  time = 0.02, antiderivative size = 53, 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 {1}{128 x^2}+\frac {9}{128 x}+\frac {27}{128 (3 x+2)}+\frac {9}{128 (3 x+2)^2}+\frac {27 \log (x)}{128}-\frac {27}{128} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(4 + 6*x)^3),x]

[Out]

-1/(128*x^2) + 9/(128*x) + 9/(128*(2 + 3*x)^2) + 27/(128*(2 + 3*x)) + (27*Log[x])/128 - (27*Log[2 + 3*x])/128

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^3 (4+6 x)^3} \, dx &=\int \left (\frac {1}{64 x^3}-\frac {9}{128 x^2}+\frac {27}{128 x}-\frac {27}{64 (2+3 x)^3}-\frac {81}{128 (2+3 x)^2}-\frac {81}{128 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{128 x^2}+\frac {9}{128 x}+\frac {9}{128 (2+3 x)^2}+\frac {27}{128 (2+3 x)}+\frac {27 \log (x)}{128}-\frac {27}{128} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 44, normalized size = 0.83 \begin {gather*} \frac {1}{128} \left (\frac {2 \left (81 x^3+81 x^2+12 x-2\right )}{x^2 (3 x+2)^2}+27 \log (x)-27 \log (3 x+2)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(4 + 6*x)^3),x]

[Out]

((2*(-2 + 12*x + 81*x^2 + 81*x^3))/(x^2*(2 + 3*x)^2) + 27*Log[x] - 27*Log[2 + 3*x])/128

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.95, size = 79, normalized size = 1.49 \begin {gather*} \frac {162 \, x^{3} + 162 \, x^{2} - 27 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )} \log \left (3 \, x + 2\right ) + 27 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )} \log \relax (x) + 24 \, x - 4}{128 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(162*x^3 + 162*x^2 - 27*(9*x^4 + 12*x^3 + 4*x^2)*log(3*x + 2) + 27*(9*x^4 + 12*x^3 + 4*x^2)*log(x) + 24*
x - 4)/(9*x^4 + 12*x^3 + 4*x^2)

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giac [A]  time = 1.07, size = 43, normalized size = 0.81 \begin {gather*} \frac {81 \, x^{3} + 81 \, x^{2} + 12 \, x - 2}{64 \, {\left (3 \, x^{2} + 2 \, x\right )}^{2}} - \frac {27}{128} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {27}{128} \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(81*x^3 + 81*x^2 + 12*x - 2)/(3*x^2 + 2*x)^2 - 27/128*log(abs(3*x + 2)) + 27/128*log(abs(x))

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maple [A]  time = 0.01, size = 42, normalized size = 0.79 \begin {gather*} \frac {27 \ln \relax (x )}{128}-\frac {27 \ln \left (3 x +2\right )}{128}+\frac {9}{128 x}-\frac {1}{128 x^{2}}+\frac {9}{128 \left (3 x +2\right )^{2}}+\frac {27}{128 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128/x^2+9/128/x+9/128/(3*x+2)^2+27/128/(3*x+2)+27/128*ln(x)-27/128*ln(3*x+2)

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maxima [A]  time = 1.38, size = 48, normalized size = 0.91 \begin {gather*} \frac {81 \, x^{3} + 81 \, x^{2} + 12 \, x - 2}{64 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )}} - \frac {27}{128} \, \log \left (3 \, x + 2\right ) + \frac {27}{128} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(81*x^3 + 81*x^2 + 12*x - 2)/(9*x^4 + 12*x^3 + 4*x^2) - 27/128*log(3*x + 2) + 27/128*log(x)

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mupad [B]  time = 0.09, size = 41, normalized size = 0.77 \begin {gather*} \frac {\frac {9\,x^3}{64}+\frac {9\,x^2}{64}+\frac {x}{48}-\frac {1}{288}}{x^4+\frac {4\,x^3}{3}+\frac {4\,x^2}{9}}-\frac {27\,\mathrm {atanh}\left (3\,x+1\right )}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(6*x + 4)^3),x)

[Out]

(x/48 + (9*x^2)/64 + (9*x^3)/64 - 1/288)/((4*x^2)/9 + (4*x^3)/3 + x^4) - (27*atanh(3*x + 1))/64

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sympy [A]  time = 0.18, size = 46, normalized size = 0.87 \begin {gather*} \frac {27 \log {\relax (x )}}{128} - \frac {27 \log {\left (x + \frac {2}{3} \right )}}{128} + \frac {81 x^{3} + 81 x^{2} + 12 x - 2}{576 x^{4} + 768 x^{3} + 256 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27*log(x)/128 - 27*log(x + 2/3)/128 + (81*x**3 + 81*x**2 + 12*x - 2)/(576*x**4 + 768*x**3 + 256*x**2)

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