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

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

[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

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Rubi [A]  time = 0.0117359, antiderivative size = 45, 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} \[ -\frac{9}{32 x^2}+\frac{1}{8 x^3}-\frac{1}{16 x^4}+\frac{27}{32 x}+\frac{81 \log (x)}{64}-\frac{81}{64} \log (3 x+2) \]

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*}

Mathematica [A]  time = 0.0031462, size = 45, normalized size = 1. \[ -\frac{9}{32 x^2}+\frac{1}{8 x^3}-\frac{1}{16 x^4}+\frac{27}{32 x}+\frac{81 \log (x)}{64}-\frac{81}{64} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[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

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Maple [A]  time = 0.006, size = 34, normalized size = 0.8 \begin{align*} -{\frac{1}{16\,{x}^{4}}}+{\frac{1}{8\,{x}^{3}}}-{\frac{9}{32\,{x}^{2}}}+{\frac{27}{32\,x}}+{\frac{81\,\ln \left ( x \right ) }{64}}-{\frac{81\,\ln \left ( 2+3\,x \right ) }{64}} \end{align*}

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(2+3*x)

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Maxima [A]  time = 1.02507, size = 45, normalized size = 1. \begin{align*} \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 \left (x\right ) \end{align*}

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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Fricas [A]  time = 1.69802, size = 104, normalized size = 2.31 \begin{align*} -\frac{81 \, x^{4} \log \left (3 \, x + 2\right ) - 81 \, x^{4} \log \left (x\right ) - 54 \, x^{3} + 18 \, x^{2} - 8 \, x + 4}{64 \, x^{4}} \end{align*}

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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Sympy [A]  time = 0.193303, size = 36, normalized size = 0.8 \begin{align*} \frac{81 \log{\left (x \right )}}{64} - \frac{81 \log{\left (x + \frac{2}{3} \right )}}{64} + \frac{27 x^{3} - 9 x^{2} + 4 x - 2}{32 x^{4}} \end{align*}

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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Giac [A]  time = 1.2037, size = 47, normalized size = 1.04 \begin{align*} \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 ) \end{align*}

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))