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

Optimal. Leaf size=24 \[ -\frac{1}{4 x}-\frac{3 \log (x)}{8}+\frac{3}{8} \log (3 x+2) \]

[Out]

-1/(4*x) - (3*Log[x])/8 + (3*Log[2 + 3*x])/8

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Rubi [A]  time = 0.0078345, antiderivative size = 24, 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} \[ -\frac{1}{4 x}-\frac{3 \log (x)}{8}+\frac{3}{8} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(4 + 6*x)),x]

[Out]

-1/(4*x) - (3*Log[x])/8 + (3*Log[2 + 3*x])/8

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

Mathematica [A]  time = 0.0026833, size = 24, normalized size = 1. \[ -\frac{1}{4 x}-\frac{3 \log (x)}{8}+\frac{3}{8} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(4 + 6*x)),x]

[Out]

-1/(4*x) - (3*Log[x])/8 + (3*Log[2 + 3*x])/8

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Maple [A]  time = 0.007, size = 19, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,x}}-{\frac{3\,\ln \left ( x \right ) }{8}}+{\frac{3\,\ln \left ( 2+3\,x \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/x-3/8*ln(x)+3/8*ln(2+3*x)

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Maxima [A]  time = 1.03344, size = 24, normalized size = 1. \begin{align*} -\frac{1}{4 \, x} + \frac{3}{8} \, \log \left (3 \, x + 2\right ) - \frac{3}{8} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/x + 3/8*log(3*x + 2) - 3/8*log(x)

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Fricas [A]  time = 1.7301, size = 58, normalized size = 2.42 \begin{align*} \frac{3 \, x \log \left (3 \, x + 2\right ) - 3 \, x \log \left (x\right ) - 2}{8 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.123414, size = 20, normalized size = 0.83 \begin{align*} - \frac{3 \log{\left (x \right )}}{8} + \frac{3 \log{\left (x + \frac{2}{3} \right )}}{8} - \frac{1}{4 x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*log(x)/8 + 3*log(x + 2/3)/8 - 1/(4*x)

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Giac [A]  time = 1.17741, size = 27, normalized size = 1.12 \begin{align*} -\frac{1}{4 \, x} + \frac{3}{8} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{3}{8} \, \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),x, algorithm="giac")

[Out]

-1/4/x + 3/8*log(abs(3*x + 2)) - 3/8*log(abs(x))