3.260 \(\int \frac {1}{x^5 (4+6 x)} \, dx\)

Optimal. Leaf size=45 \[ -\frac {1}{16 x^4}+\frac {1}{8 x^3}-\frac {9}{32 x^2}+\frac {27}{32 x}+\frac {81 \log (x)}{64}-\frac {81}{64} \log (3 x+2) \]

[Out]

-1/16/x^4+1/8/x^3-9/32/x^2+27/32/x+81/64*ln(x)-81/64*ln(2+3*x)

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Rubi [A]  time = 0.01, antiderivative size = 45, 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} \[ -\frac {9}{32 x^2}+\frac {1}{8 x^3}-\frac {1}{16 x^4}+\frac {27}{32 x}+\frac {81 \log (x)}{64}-\frac {81}{64} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(4 + 6*x)),x]

[Out]

-1/(16*x^4) + 1/(8*x^3) - 9/(32*x^2) + 27/(32*x) + (81*Log[x])/64 - (81*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^5 (4+6 x)} \, dx &=\int \left (\frac {1}{4 x^5}-\frac {3}{8 x^4}+\frac {9}{16 x^3}-\frac {27}{32 x^2}+\frac {81}{64 x}-\frac {243}{64 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{16 x^4}+\frac {1}{8 x^3}-\frac {9}{32 x^2}+\frac {27}{32 x}+\frac {81 \log (x)}{64}-\frac {81}{64} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 45, normalized size = 1.00 \[ -\frac {1}{16 x^4}+\frac {1}{8 x^3}-\frac {9}{32 x^2}+\frac {27}{32 x}+\frac {81 \log (x)}{64}-\frac {81}{64} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(4 + 6*x)),x]

[Out]

-1/16*1/x^4 + 1/(8*x^3) - 9/(32*x^2) + 27/(32*x) + (81*Log[x])/64 - (81*Log[2 + 3*x])/64

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fricas [A]  time = 0.48, size = 38, normalized size = 0.84 \[ -\frac {81 \, x^{4} \log \left (3 \, x + 2\right ) - 81 \, x^{4} \log \relax (x) - 54 \, x^{3} + 18 \, x^{2} - 8 \, x + 4}{64 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(4+6*x),x, algorithm="fricas")

[Out]

-1/64*(81*x^4*log(3*x + 2) - 81*x^4*log(x) - 54*x^3 + 18*x^2 - 8*x + 4)/x^4

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giac [A]  time = 1.14, size = 35, normalized size = 0.78 \[ \frac {27 \, x^{3} - 9 \, x^{2} + 4 \, x - 2}{32 \, x^{4}} - \frac {81}{64} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {81}{64} \, \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(4+6*x),x, algorithm="giac")

[Out]

1/32*(27*x^3 - 9*x^2 + 4*x - 2)/x^4 - 81/64*log(abs(3*x + 2)) + 81/64*log(abs(x))

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maple [A]  time = 0.01, size = 34, normalized size = 0.76 \[ \frac {81 \ln \relax (x )}{64}-\frac {81 \ln \left (3 x +2\right )}{64}+\frac {27}{32 x}-\frac {9}{32 x^{2}}+\frac {1}{8 x^{3}}-\frac {1}{16 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(4+6*x),x)

[Out]

-1/16/x^4+1/8/x^3-9/32/x^2+27/32/x+81/64*ln(x)-81/64*ln(3*x+2)

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maxima [A]  time = 1.32, size = 33, normalized size = 0.73 \[ \frac {27 \, x^{3} - 9 \, x^{2} + 4 \, x - 2}{32 \, x^{4}} - \frac {81}{64} \, \log \left (3 \, x + 2\right ) + \frac {81}{64} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(4+6*x),x, algorithm="maxima")

[Out]

1/32*(27*x^3 - 9*x^2 + 4*x - 2)/x^4 - 81/64*log(3*x + 2) + 81/64*log(x)

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mupad [B]  time = 0.04, size = 28, normalized size = 0.62 \[ \frac {\frac {27\,x^3}{32}-\frac {9\,x^2}{32}+\frac {x}{8}-\frac {1}{16}}{x^4}-\frac {81\,\mathrm {atanh}\left (3\,x+1\right )}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(6*x + 4)),x)

[Out]

(x/8 - (9*x^2)/32 + (27*x^3)/32 - 1/16)/x^4 - (81*atanh(3*x + 1))/32

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sympy [A]  time = 0.17, size = 36, normalized size = 0.80 \[ \frac {81 \log {\relax (x )}}{64} - \frac {81 \log {\left (x + \frac {2}{3} \right )}}{64} + \frac {27 x^{3} - 9 x^{2} + 4 x - 2}{32 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(4+6*x),x)

[Out]

81*log(x)/64 - 81*log(x + 2/3)/64 + (27*x**3 - 9*x**2 + 4*x - 2)/(32*x**4)

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