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

Optimal. Leaf size=38 \[ \frac{3}{16 x^2}-\frac{1}{12 x^3}-\frac{9}{16 x}-\frac{27 \log (x)}{32}+\frac{27}{32} \log (3 x+2) \]

[Out]

-1/(12*x^3) + 3/(16*x^2) - 9/(16*x) - (27*Log[x])/32 + (27*Log[2 + 3*x])/32

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Rubi [A]  time = 0.0117281, antiderivative size = 38, 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} \[ \frac{3}{16 x^2}-\frac{1}{12 x^3}-\frac{9}{16 x}-\frac{27 \log (x)}{32}+\frac{27}{32} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(4 + 6*x)),x]

[Out]

-1/(12*x^3) + 3/(16*x^2) - 9/(16*x) - (27*Log[x])/32 + (27*Log[2 + 3*x])/32

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 (4+6 x)} \, dx &=\int \left (\frac{1}{4 x^4}-\frac{3}{8 x^3}+\frac{9}{16 x^2}-\frac{27}{32 x}+\frac{81}{32 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{12 x^3}+\frac{3}{16 x^2}-\frac{9}{16 x}-\frac{27 \log (x)}{32}+\frac{27}{32} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0065298, size = 38, normalized size = 1. \[ \frac{3}{16 x^2}-\frac{1}{12 x^3}-\frac{9}{16 x}-\frac{27 \log (x)}{32}+\frac{27}{32} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(4 + 6*x)),x]

[Out]

-1/(12*x^3) + 3/(16*x^2) - 9/(16*x) - (27*Log[x])/32 + (27*Log[2 + 3*x])/32

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Maple [A]  time = 0.005, size = 29, normalized size = 0.8 \begin{align*} -{\frac{1}{12\,{x}^{3}}}+{\frac{3}{16\,{x}^{2}}}-{\frac{9}{16\,x}}-{\frac{27\,\ln \left ( x \right ) }{32}}+{\frac{27\,\ln \left ( 2+3\,x \right ) }{32}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12/x^3+3/16/x^2-9/16/x-27/32*ln(x)+27/32*ln(2+3*x)

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Maxima [A]  time = 1.03935, size = 38, normalized size = 1. \begin{align*} -\frac{27 \, x^{2} - 9 \, x + 4}{48 \, x^{3}} + \frac{27}{32} \, \log \left (3 \, x + 2\right ) - \frac{27}{32} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(27*x^2 - 9*x + 4)/x^3 + 27/32*log(3*x + 2) - 27/32*log(x)

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Fricas [A]  time = 1.79884, size = 92, normalized size = 2.42 \begin{align*} \frac{81 \, x^{3} \log \left (3 \, x + 2\right ) - 81 \, x^{3} \log \left (x\right ) - 54 \, x^{2} + 18 \, x - 8}{96 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(81*x^3*log(3*x + 2) - 81*x^3*log(x) - 54*x^2 + 18*x - 8)/x^3

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Sympy [A]  time = 0.21336, size = 31, normalized size = 0.82 \begin{align*} - \frac{27 \log{\left (x \right )}}{32} + \frac{27 \log{\left (x + \frac{2}{3} \right )}}{32} - \frac{27 x^{2} - 9 x + 4}{48 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27*log(x)/32 + 27*log(x + 2/3)/32 - (27*x**2 - 9*x + 4)/(48*x**3)

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Giac [A]  time = 1.14213, size = 41, normalized size = 1.08 \begin{align*} -\frac{27 \, x^{2} - 9 \, x + 4}{48 \, x^{3}} + \frac{27}{32} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{27}{32} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(27*x^2 - 9*x + 4)/x^3 + 27/32*log(abs(3*x + 2)) - 27/32*log(abs(x))