3.267 \(\int \frac{1}{x^2 (4+6 x)^3} \, dx\)

Optimal. Leaf size=46 \[ -\frac{1}{64 x}-\frac{3}{32 (3 x+2)}-\frac{3}{64 (3 x+2)^2}-\frac{9 \log (x)}{128}+\frac{9}{128} \log (3 x+2) \]

[Out]

-1/(64*x) - 3/(64*(2 + 3*x)^2) - 3/(32*(2 + 3*x)) - (9*Log[x])/128 + (9*Log[2 + 3*x])/128

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Rubi [A]  time = 0.0150952, antiderivative size = 46, 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} \[ -\frac{1}{64 x}-\frac{3}{32 (3 x+2)}-\frac{3}{64 (3 x+2)^2}-\frac{9 \log (x)}{128}+\frac{9}{128} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(4 + 6*x)^3),x]

[Out]

-1/(64*x) - 3/(64*(2 + 3*x)^2) - 3/(32*(2 + 3*x)) - (9*Log[x])/128 + (9*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^2 (4+6 x)^3} \, dx &=\int \left (\frac{1}{64 x^2}-\frac{9}{128 x}+\frac{9}{32 (2+3 x)^3}+\frac{9}{32 (2+3 x)^2}+\frac{27}{128 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{64 x}-\frac{3}{64 (2+3 x)^2}-\frac{3}{32 (2+3 x)}-\frac{9 \log (x)}{128}+\frac{9}{128} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0343409, size = 39, normalized size = 0.85 \[ \frac{1}{128} \left (-\frac{2 \left (27 x^2+27 x+4\right )}{x (3 x+2)^2}-9 \log (x)+9 \log (3 x+2)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(4 + 6*x)^3),x]

[Out]

((-2*(4 + 27*x + 27*x^2))/(x*(2 + 3*x)^2) - 9*Log[x] + 9*Log[2 + 3*x])/128

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Maple [A]  time = 0.009, size = 37, normalized size = 0.8 \begin{align*} -{\frac{1}{64\,x}}-{\frac{3}{64\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{3}{64+96\,x}}-{\frac{9\,\ln \left ( x \right ) }{128}}+{\frac{9\,\ln \left ( 2+3\,x \right ) }{128}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(4+6*x)^3,x)

[Out]

-1/64/x-3/64/(2+3*x)^2-3/32/(2+3*x)-9/128*ln(x)+9/128*ln(2+3*x)

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Maxima [A]  time = 1.06005, size = 55, normalized size = 1.2 \begin{align*} -\frac{27 \, x^{2} + 27 \, x + 4}{64 \,{\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )}} + \frac{9}{128} \, \log \left (3 \, x + 2\right ) - \frac{9}{128} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(4+6*x)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.76173, size = 169, normalized size = 3.67 \begin{align*} -\frac{54 \, x^{2} - 9 \,{\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )} \log \left (3 \, x + 2\right ) + 9 \,{\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )} \log \left (x\right ) + 54 \, x + 8}{128 \,{\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(4+6*x)^3,x, algorithm="fricas")

[Out]

-1/128*(54*x^2 - 9*(9*x^3 + 12*x^2 + 4*x)*log(3*x + 2) + 9*(9*x^3 + 12*x^2 + 4*x)*log(x) + 54*x + 8)/(9*x^3 +
12*x^2 + 4*x)

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Sympy [A]  time = 0.16527, size = 39, normalized size = 0.85 \begin{align*} - \frac{27 x^{2} + 27 x + 4}{576 x^{3} + 768 x^{2} + 256 x} - \frac{9 \log{\left (x \right )}}{128} + \frac{9 \log{\left (x + \frac{2}{3} \right )}}{128} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(4+6*x)**3,x)

[Out]

-(27*x**2 + 27*x + 4)/(576*x**3 + 768*x**2 + 256*x) - 9*log(x)/128 + 9*log(x + 2/3)/128

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Giac [A]  time = 1.23613, size = 50, normalized size = 1.09 \begin{align*} -\frac{27 \, x^{2} + 27 \, x + 4}{64 \,{\left (3 \, x + 2\right )}^{2} x} + \frac{9}{128} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{9}{128} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(4+6*x)^3,x, algorithm="giac")

[Out]

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