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

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.015826, antiderivative size = 53, 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} \[ -\frac{1}{128 x^2}+\frac{9}{128 x}+\frac{27}{128 (3 x+2)}+\frac{9}{128 (3 x+2)^2}+\frac{27 \log (x)}{128}-\frac{27}{128} \log (3 x+2) \]

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*}

Mathematica [A]  time = 0.0369721, size = 44, normalized size = 0.83 \[ \frac{1}{128} \left (\frac{2 \left (81 x^3+81 x^2+12 x-2\right )}{x^2 (3 x+2)^2}+27 \log (x)-27 \log (3 x+2)\right ) \]

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

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 42, normalized size = 0.8 \begin{align*} -{\frac{1}{128\,{x}^{2}}}+{\frac{9}{128\,x}}+{\frac{9}{128\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{27}{256+384\,x}}+{\frac{27\,\ln \left ( x \right ) }{128}}-{\frac{27\,\ln \left ( 2+3\,x \right ) }{128}} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.0443, size = 65, normalized size = 1.23 \begin{align*} \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 \left (x\right ) \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.80194, size = 193, normalized size = 3.64 \begin{align*} \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 \left (x\right ) + 24 \, x - 4}{128 \,{\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )}} \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0.239392, size = 46, normalized size = 0.87 \begin{align*} \frac{27 \log{\left (x \right )}}{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{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.23302, size = 58, normalized size = 1.09 \begin{align*} \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{align*}

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))