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

Optimal. Leaf size=31 \[ -\frac{1}{8 x^2}+\frac{3}{8 x}+\frac{9 \log (x)}{16}-\frac{9}{16} \log (3 x+2) \]

[Out]

-1/(8*x^2) + 3/(8*x) + (9*Log[x])/16 - (9*Log[2 + 3*x])/16

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Rubi [A]  time = 0.0100174, antiderivative size = 31, 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} \[ -\frac{1}{8 x^2}+\frac{3}{8 x}+\frac{9 \log (x)}{16}-\frac{9}{16} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(4 + 6*x)),x]

[Out]

-1/(8*x^2) + 3/(8*x) + (9*Log[x])/16 - (9*Log[2 + 3*x])/16

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)} \, dx &=\int \left (\frac{1}{4 x^3}-\frac{3}{8 x^2}+\frac{9}{16 x}-\frac{27}{16 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{8 x^2}+\frac{3}{8 x}+\frac{9 \log (x)}{16}-\frac{9}{16} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0028378, size = 31, normalized size = 1. \[ -\frac{1}{8 x^2}+\frac{3}{8 x}+\frac{9 \log (x)}{16}-\frac{9}{16} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(4 + 6*x)),x]

[Out]

-1/(8*x^2) + 3/(8*x) + (9*Log[x])/16 - (9*Log[2 + 3*x])/16

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Maple [A]  time = 0.007, size = 24, normalized size = 0.8 \begin{align*} -{\frac{1}{8\,{x}^{2}}}+{\frac{3}{8\,x}}+{\frac{9\,\ln \left ( x \right ) }{16}}-{\frac{9\,\ln \left ( 2+3\,x \right ) }{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/x^2+3/8/x+9/16*ln(x)-9/16*ln(2+3*x)

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Maxima [A]  time = 0.991965, size = 31, normalized size = 1. \begin{align*} \frac{3 \, x - 1}{8 \, x^{2}} - \frac{9}{16} \, \log \left (3 \, x + 2\right ) + \frac{9}{16} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*x - 1)/x^2 - 9/16*log(3*x + 2) + 9/16*log(x)

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Fricas [A]  time = 1.74511, size = 77, normalized size = 2.48 \begin{align*} -\frac{9 \, x^{2} \log \left (3 \, x + 2\right ) - 9 \, x^{2} \log \left (x\right ) - 6 \, x + 2}{16 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(9*x^2*log(3*x + 2) - 9*x^2*log(x) - 6*x + 2)/x^2

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Sympy [A]  time = 0.167408, size = 26, normalized size = 0.84 \begin{align*} \frac{9 \log{\left (x \right )}}{16} - \frac{9 \log{\left (x + \frac{2}{3} \right )}}{16} + \frac{3 x - 1}{8 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9*log(x)/16 - 9*log(x + 2/3)/16 + (3*x - 1)/(8*x**2)

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Giac [A]  time = 1.16679, size = 34, normalized size = 1.1 \begin{align*} \frac{3 \, x - 1}{8 \, x^{2}} - \frac{9}{16} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac{9}{16} \, \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),x, algorithm="giac")

[Out]

1/8*(3*x - 1)/x^2 - 9/16*log(abs(3*x + 2)) + 9/16*log(abs(x))