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

Optimal. Leaf size=28 \[ \frac{1}{8 (3 x+2)}+\frac{\log (x)}{16}-\frac{1}{16} \log (3 x+2) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0089128, antiderivative size = 28, 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} \[ \frac{1}{8 (3 x+2)}+\frac{\log (x)}{16}-\frac{1}{16} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(4 + 6*x)^2),x]

[Out]

1/(8*(2 + 3*x)) + Log[x]/16 - 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 (4+6 x)^2} \, dx &=\int \left (\frac{1}{16 x}-\frac{3}{8 (2+3 x)^2}-\frac{3}{16 (2+3 x)}\right ) \, dx\\ &=\frac{1}{8 (2+3 x)}+\frac{\log (x)}{16}-\frac{1}{16} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0226824, size = 26, normalized size = 0.93 \[ \frac{1}{16} \left (\frac{2}{3 x+2}+\log (-6 x)-\log (6 x+4)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(4 + 6*x)^2),x]

[Out]

(2/(2 + 3*x) + Log[-6*x] - Log[4 + 6*x])/16

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 23, normalized size = 0.8 \begin{align*}{\frac{1}{16+24\,x}}+{\frac{\ln \left ( x \right ) }{16}}-{\frac{\ln \left ( 2+3\,x \right ) }{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.08183, size = 30, normalized size = 1.07 \begin{align*} \frac{1}{8 \,{\left (3 \, x + 2\right )}} - \frac{1}{16} \, \log \left (3 \, x + 2\right ) + \frac{1}{16} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.68688, size = 88, normalized size = 3.14 \begin{align*} -\frac{{\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) -{\left (3 \, x + 2\right )} \log \left (x\right ) - 2}{16 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.113678, size = 19, normalized size = 0.68 \begin{align*} \frac{\log{\left (x \right )}}{16} - \frac{\log{\left (x + \frac{2}{3} \right )}}{16} + \frac{1}{24 x + 16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/16 - log(x + 2/3)/16 + 1/(24*x + 16)

________________________________________________________________________________________

Giac [A]  time = 1.19607, size = 34, normalized size = 1.21 \begin{align*} \frac{1}{8 \,{\left (3 \, x + 2\right )}} + \frac{1}{16} \, \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/(4+6*x)^2,x, algorithm="giac")

[Out]

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