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

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

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Rubi [A]  time = 0.01, antiderivative size = 38, 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 {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) \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 38, normalized size = 1.00 \begin {gather*} -\frac {1}{12 x^3}+\frac {3}{16 x^2}-\frac {9}{16 x}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(4 + 6*x)),x]

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.57, size = 33, normalized size = 0.87 \begin {gather*} \frac {81 \, x^{3} \log \left (3 \, x + 2\right ) - 81 \, x^{3} \log \relax (x) - 54 \, x^{2} + 18 \, x - 8}{96 \, x^{3}} \end {gather*}

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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giac [A]  time = 1.03, size = 30, normalized size = 0.79 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.01, size = 29, normalized size = 0.76 \begin {gather*} -\frac {27 \ln \relax (x )}{32}+\frac {27 \ln \left (3 x +2\right )}{32}-\frac {9}{16 x}+\frac {3}{16 x^{2}}-\frac {1}{12 x^{3}} \end {gather*}

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(3*x+2)

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maxima [A]  time = 1.35, size = 28, normalized size = 0.74 \begin {gather*} -\frac {27 \, x^{2} - 9 \, x + 4}{48 \, x^{3}} + \frac {27}{32} \, \log \left (3 \, x + 2\right ) - \frac {27}{32} \, \log \relax (x) \end {gather*}

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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mupad [B]  time = 0.09, size = 24, normalized size = 0.63 \begin {gather*} \frac {27\,\mathrm {atanh}\left (3\,x+1\right )}{16}-\frac {\frac {9\,x^2}{16}-\frac {3\,x}{16}+\frac {1}{12}}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(6*x + 4)),x)

[Out]

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

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sympy [A]  time = 0.16, size = 31, normalized size = 0.82 \begin {gather*} - \frac {27 \log {\relax (x )}}{32} + \frac {27 \log {\left (x + \frac {2}{3} \right )}}{32} + \frac {- 27 x^{2} + 9 x - 4}{48 x^{3}} \end {gather*}

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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