3.3.64 \(\int \frac {1}{x^4 (4+6 x)^2} \, dx\) [264]

Optimal. Leaf size=49 \[ -\frac {1}{48 x^3}+\frac {3}{32 x^2}-\frac {27}{64 x}-\frac {27}{64 (2+3 x)}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (2+3 x) \]

[Out]

-1/48/x^3+3/32/x^2-27/64/x-27/64/(2+3*x)-27/32*ln(x)+27/32*ln(2+3*x)

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Rubi [A]
time = 0.01, antiderivative size = 49, 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}{48 x^3}+\frac {3}{32 x^2}-\frac {27}{64 x}-\frac {27}{64 (3 x+2)}-\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)^2),x]

[Out]

-1/48*1/x^3 + 3/(32*x^2) - 27/(64*x) - 27/(64*(2 + 3*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)^2} \, dx &=\int \left (\frac {1}{16 x^4}-\frac {3}{16 x^3}+\frac {27}{64 x^2}-\frac {27}{32 x}+\frac {81}{64 (2+3 x)^2}+\frac {81}{32 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{48 x^3}+\frac {3}{32 x^2}-\frac {27}{64 x}-\frac {27}{64 (2+3 x)}-\frac {27 \log (x)}{32}+\frac {27}{32} \log (2+3 x)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 44, normalized size = 0.90 \begin {gather*} \frac {1}{192} \left (-\frac {4 \left (2-6 x+27 x^2+81 x^3\right )}{x^3 (2+3 x)}-162 \log (x)+162 \log (2+3 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(4 + 6*x)^2),x]

[Out]

((-4*(2 - 6*x + 27*x^2 + 81*x^3))/(x^3*(2 + 3*x)) - 162*Log[x] + 162*Log[2 + 3*x])/192

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Mathics [A]
time = 1.87, size = 46, normalized size = 0.94 \begin {gather*} \frac {-4+12 x-54 x^2-162 x^3+81 x^3 \left (2+3 x\right ) \left (\text {Log}\left [\frac {2}{3}+x\right ]-\text {Log}\left [x\right ]\right )}{96 x^3 \left (2+3 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(x^4*(4 + 6*x)^2),x]')

[Out]

(-4 + 12 x - 54 x ^ 2 - 162 x ^ 3 + 81 x ^ 3 (2 + 3 x) (Log[2 / 3 + x] - Log[x])) / (96 x ^ 3 (2 + 3 x))

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Maple [A]
time = 0.10, size = 38, normalized size = 0.78

method	result	size
default	\(-\frac {1}{48 x^{3}}+\frac {3}{32 x^{2}}-\frac {27}{64 x}-\frac {27}{64 \left (2+3 x \right )}-\frac {27 \ln \left (x \right )}{32}+\frac {27 \ln \left (2+3 x \right )}{32}\)	\(38\)
norman	\(\frac {-\frac {1}{24}+\frac {81}{32} x^{4}+\frac {1}{8} x -\frac {9}{16} x^{2}}{x^{3} \left (2+3 x \right )}-\frac {27 \ln \left (x \right )}{32}+\frac {27 \ln \left (2+3 x \right )}{32}\)	\(40\)
risch	\(\frac {-\frac {27}{16} x^{3}-\frac {9}{16} x^{2}+\frac {1}{8} x -\frac {1}{24}}{x^{3} \left (2+3 x \right )}-\frac {27 \ln \left (x \right )}{32}+\frac {27 \ln \left (2+3 x \right )}{32}\)	\(41\)
meijerg	\(-\frac {1}{48 x^{3}}+\frac {3}{32 x^{2}}-\frac {27}{64 x}-\frac {27}{128}-\frac {27 \ln \left (x \right )}{32}+\frac {27 \ln \left (2\right )}{32}-\frac {27 \ln \left (3\right )}{32}+\frac {405 x}{256 \left (5+\frac {15 x}{2}\right )}+\frac {27 \ln \left (1+\frac {3 x}{2}\right )}{32}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(4+6*x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/48/x^3+3/32/x^2-27/64/x-27/64/(2+3*x)-27/32*ln(x)+27/32*ln(2+3*x)

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Maxima [A]
time = 0.25, size = 43, normalized size = 0.88 \begin {gather*} -\frac {81 \, x^{3} + 27 \, x^{2} - 6 \, x + 2}{48 \, {\left (3 \, x^{4} + 2 \, x^{3}\right )}} + \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)^2,x, algorithm="maxima")

[Out]

-1/48*(81*x^3 + 27*x^2 - 6*x + 2)/(3*x^4 + 2*x^3) + 27/32*log(3*x + 2) - 27/32*log(x)

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Fricas [A]
time = 0.31, size = 64, normalized size = 1.31 \begin {gather*} -\frac {162 \, x^{3} + 54 \, x^{2} - 81 \, {\left (3 \, x^{4} + 2 \, x^{3}\right )} \log \left (3 \, x + 2\right ) + 81 \, {\left (3 \, x^{4} + 2 \, x^{3}\right )} \log \left (x\right ) - 12 \, x + 4}{96 \, {\left (3 \, x^{4} + 2 \, x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(4+6*x)^2,x, algorithm="fricas")

[Out]

-1/96*(162*x^3 + 54*x^2 - 81*(3*x^4 + 2*x^3)*log(3*x + 2) + 81*(3*x^4 + 2*x^3)*log(x) - 12*x + 4)/(3*x^4 + 2*x
^3)

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Sympy [A]
time = 0.07, size = 41, normalized size = 0.84 \begin {gather*} - \frac {27 \log {\left (x \right )}}{32} + \frac {27 \log {\left (x + \frac {2}{3} \right )}}{32} + \frac {- 81 x^{3} - 27 x^{2} + 6 x - 2}{144 x^{4} + 96 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(4+6*x)**2,x)

[Out]

-27*log(x)/32 + 27*log(x + 2/3)/32 + (-81*x**3 - 27*x**2 + 6*x - 2)/(144*x**4 + 96*x**3)

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Giac [A]
time = 0.00, size = 50, normalized size = 1.02 \begin {gather*} \frac {27}{32} \ln \left |3 x+2\right |-\frac {27}{32} \ln \left |x\right |+\frac {\frac {1}{192} \left (-324 x^{3}-108 x^{2}+24 x-8\right )}{x^{3} \left (3 x+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(4+6*x)^2,x)

[Out]

-1/48*(81*x^3 + 27*x^2 - 6*x + 2)/((3*x + 2)*x^3) + 27/32*log(abs(3*x + 2)) - 27/32*log(abs(x))

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Mupad [B]
time = 0.09, size = 37, normalized size = 0.76 \begin {gather*} \frac {27\,\mathrm {atanh}\left (3\,x+1\right )}{16}-\frac {\frac {9\,x^3}{16}+\frac {3\,x^2}{16}-\frac {x}{24}+\frac {1}{72}}{x^4+\frac {2\,x^3}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(6*x + 4)^2),x)

[Out]

(27*atanh(3*x + 1))/16 - ((3*x^2)/16 - x/24 + (9*x^3)/16 + 1/72)/((2*x^3)/3 + x^4)

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