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

Optimal. Leaf size=39 \[ \frac {1}{32 (3 x+2)}+\frac {1}{32 (3 x+2)^2}+\frac {\log (x)}{64}-\frac {1}{64} \log (3 x+2) \]

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Rubi [A]  time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{32 (3 x+2)}+\frac {1}{32 (3 x+2)^2}+\frac {\log (x)}{64}-\frac {1}{64} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(4 + 6*x)^3),x]

[Out]

1/(32*(2 + 3*x)^2) + 1/(32*(2 + 3*x)) + Log[x]/64 - 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 (4+6 x)^3} \, dx &=\int \left (\frac {1}{64 x}-\frac {3}{16 (2+3 x)^3}-\frac {3}{32 (2+3 x)^2}-\frac {3}{64 (2+3 x)}\right ) \, dx\\ &=\frac {1}{32 (2+3 x)^2}+\frac {1}{32 (2+3 x)}+\frac {\log (x)}{64}-\frac {1}{64} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 29, normalized size = 0.74 \begin {gather*} \frac {1}{64} \left (\frac {6 (x+1)}{(3 x+2)^2}+\log (-6 x)-\log (6 x+4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(4 + 6*x)^3),x]

[Out]

((6*(1 + x))/(2 + 3*x)^2 + Log[-6*x] - Log[4 + 6*x])/64

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x (4+6 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x*(4 + 6*x)^3),x]

[Out]

IntegrateAlgebraic[1/(x*(4 + 6*x)^3), x]

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fricas [A]  time = 1.08, size = 50, normalized size = 1.28 \begin {gather*} -\frac {{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) - {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \relax (x) - 6 \, x - 6}{64 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*((9*x^2 + 12*x + 4)*log(3*x + 2) - (9*x^2 + 12*x + 4)*log(x) - 6*x - 6)/(9*x^2 + 12*x + 4)

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giac [A]  time = 1.03, size = 27, normalized size = 0.69 \begin {gather*} \frac {3 \, {\left (x + 1\right )}}{32 \, {\left (3 \, x + 2\right )}^{2}} - \frac {1}{64} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {1}{64} \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 32, normalized size = 0.82 \begin {gather*} \frac {\ln \relax (x )}{64}-\frac {\ln \left (3 x +2\right )}{64}+\frac {1}{32 \left (3 x +2\right )^{2}}+\frac {1}{96 x +64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.39, size = 30, normalized size = 0.77 \begin {gather*} \frac {3 \, {\left (x + 1\right )}}{32 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1}{64} \, \log \left (3 \, x + 2\right ) + \frac {1}{64} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/32*(x + 1)/(9*x^2 + 12*x + 4) - 1/64*log(3*x + 2) + 1/64*log(x)

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mupad [B]  time = 0.13, size = 29, normalized size = 0.74 \begin {gather*} \frac {1}{32\,\left (3\,x+2\right )}-\frac {\ln \left (\frac {6\,x+4}{x}\right )}{64}+\frac {1}{8\,{\left (6\,x+4\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(6*x + 4)^3),x)

[Out]

1/(32*(3*x + 2)) - log((6*x + 4)/x)/64 + 1/(8*(6*x + 4)^2)

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sympy [A]  time = 0.17, size = 27, normalized size = 0.69 \begin {gather*} \frac {3 x + 3}{288 x^{2} + 384 x + 128} + \frac {\log {\relax (x )}}{64} - \frac {\log {\left (x + \frac {2}{3} \right )}}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*x + 3)/(288*x**2 + 384*x + 128) + log(x)/64 - log(x + 2/3)/64

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