3.3.59 \(\int \frac {1}{x^4 (4+6 x)} \, dx\) [259]

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

[Out]

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

________________________________________________________________________________________

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 = {46} \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]

Int[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

Rule 46

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] && Lt
Q[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 = 0.00, 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 (2+3 x) \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

________________________________________________________________________________________

Mathics [A]
time = 1.74, size = 29, normalized size = 0.76 \begin {gather*} \frac {-8+18 x-54 x^2+81 x^3 \left (\text {Log}\left [\frac {2}{3}+x\right ]-\text {Log}\left [x\right ]\right )}{96 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(x^4*(4 + 6*x)),x]')

[Out]

(-8 + 18 x - 54 x ^ 2 + 81 x ^ 3 (Log[2 / 3 + x] - Log[x])) / (96 x ^ 3)

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 29, normalized size = 0.76

method	result	size
norman	\(\frac {-\frac {1}{12}+\frac {3}{16} x -\frac {9}{16} x^{2}}{x^{3}}-\frac {27 \ln \left (x \right )}{32}+\frac {27 \ln \left (2+3 x \right )}{32}\)	\(28\)
risch	\(\frac {-\frac {1}{12}+\frac {3}{16} x -\frac {9}{16} x^{2}}{x^{3}}-\frac {27 \ln \left (x \right )}{32}+\frac {27 \ln \left (2+3 x \right )}{32}\)	\(28\)
default	\(-\frac {1}{12 x^{3}}+\frac {3}{16 x^{2}}-\frac {9}{16 x}-\frac {27 \ln \left (x \right )}{32}+\frac {27 \ln \left (2+3 x \right )}{32}\)	\(29\)
meijerg	\(-\frac {1}{12 x^{3}}+\frac {3}{16 x^{2}}-\frac {9}{16 x}-\frac {27 \ln \left (x \right )}{32}+\frac {27 \ln \left (2\right )}{32}-\frac {27 \ln \left (3\right )}{32}+\frac {27 \ln \left (1+\frac {3 x}{2}\right )}{32}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.24, 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 \left (x\right ) \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)

________________________________________________________________________________________

Fricas [A]
time = 0.31, size = 33, normalized size = 0.87 \begin {gather*} \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 {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

________________________________________________________________________________________

Sympy [A]
time = 0.07, size = 31, normalized size = 0.82 \begin {gather*} - \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 {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)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 38, normalized size = 1.00 \begin {gather*} \frac {27}{32} \ln \left |3 x+2\right |-\frac {27}{32} \ln \left |x\right |+\frac {\frac {1}{192} \left (-108 x^{2}+36 x-16\right )}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

________________________________________________________________________________________