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

Optimal. Leaf size=42 \[ -\frac{1}{32 x^2}+\frac{3}{16 x}+\frac{9}{32 (3 x+2)}+\frac{27 \log (x)}{64}-\frac{27}{64} \log (3 x+2) \]

[Out]

-1/(32*x^2) + 3/(16*x) + 9/(32*(2 + 3*x)) + (27*Log[x])/64 - (27*Log[2 + 3*x])/64

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Rubi [A]  time = 0.0135602, antiderivative size = 42, 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} \[ -\frac{1}{32 x^2}+\frac{3}{16 x}+\frac{9}{32 (3 x+2)}+\frac{27 \log (x)}{64}-\frac{27}{64} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(4 + 6*x)^2),x]

[Out]

-1/(32*x^2) + 3/(16*x) + 9/(32*(2 + 3*x)) + (27*Log[x])/64 - (27*Log[2 + 3*x])/64

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)^2} \, dx &=\int \left (\frac{1}{16 x^3}-\frac{3}{16 x^2}+\frac{27}{64 x}-\frac{27}{32 (2+3 x)^2}-\frac{81}{64 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{32 x^2}+\frac{3}{16 x}+\frac{9}{32 (2+3 x)}+\frac{27 \log (x)}{64}-\frac{27}{64} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0192639, size = 36, normalized size = 0.86 \[ \frac{1}{64} \left (-\frac{2}{x^2}+\frac{12}{x}+\frac{18}{3 x+2}+27 \log (x)-27 \log (3 x+2)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(4 + 6*x)^2),x]

[Out]

(-2/x^2 + 12/x + 18/(2 + 3*x) + 27*Log[x] - 27*Log[2 + 3*x])/64

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Maple [A]  time = 0.007, size = 33, normalized size = 0.8 \begin{align*} -{\frac{1}{32\,{x}^{2}}}+{\frac{3}{16\,x}}+{\frac{9}{64+96\,x}}+{\frac{27\,\ln \left ( x \right ) }{64}}-{\frac{27\,\ln \left ( 2+3\,x \right ) }{64}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32/x^2+3/16/x+9/32/(2+3*x)+27/64*ln(x)-27/64*ln(2+3*x)

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Maxima [A]  time = 1.08064, size = 51, normalized size = 1.21 \begin{align*} \frac{27 \, x^{2} + 9 \, x - 2}{32 \,{\left (3 \, x^{3} + 2 \, x^{2}\right )}} - \frac{27}{64} \, \log \left (3 \, x + 2\right ) + \frac{27}{64} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(27*x^2 + 9*x - 2)/(3*x^3 + 2*x^2) - 27/64*log(3*x + 2) + 27/64*log(x)

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Fricas [A]  time = 1.73493, size = 140, normalized size = 3.33 \begin{align*} \frac{54 \, x^{2} - 27 \,{\left (3 \, x^{3} + 2 \, x^{2}\right )} \log \left (3 \, x + 2\right ) + 27 \,{\left (3 \, x^{3} + 2 \, x^{2}\right )} \log \left (x\right ) + 18 \, x - 4}{64 \,{\left (3 \, x^{3} + 2 \, x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(54*x^2 - 27*(3*x^3 + 2*x^2)*log(3*x + 2) + 27*(3*x^3 + 2*x^2)*log(x) + 18*x - 4)/(3*x^3 + 2*x^2)

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Sympy [A]  time = 0.268462, size = 36, normalized size = 0.86 \begin{align*} \frac{27 \log{\left (x \right )}}{64} - \frac{27 \log{\left (x + \frac{2}{3} \right )}}{64} + \frac{27 x^{2} + 9 x - 2}{96 x^{3} + 64 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27*log(x)/64 - 27*log(x + 2/3)/64 + (27*x**2 + 9*x - 2)/(96*x**3 + 64*x**2)

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Giac [A]  time = 1.22409, size = 69, normalized size = 1.64 \begin{align*} \frac{9}{32 \,{\left (3 \, x + 2\right )}} - \frac{9 \,{\left (\frac{12}{3 \, x + 2} - 5\right )}}{128 \,{\left (\frac{2}{3 \, x + 2} - 1\right )}^{2}} + \frac{27}{64} \, \log \left ({\left | -\frac{2}{3 \, x + 2} + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/32/(3*x + 2) - 9/128*(12/(3*x + 2) - 5)/(2/(3*x + 2) - 1)^2 + 27/64*log(abs(-2/(3*x + 2) + 1))