3.3.68 \(\int \frac {1}{x^3 (4+6 x)^3} \, dx\) [268]

Optimal. Leaf size=53 \[ -\frac {1}{128 x^2}+\frac {9}{128 x}+\frac {9}{128 (2+3 x)^2}+\frac {27}{128 (2+3 x)}+\frac {27 \log (x)}{128}-\frac {27}{128} \log (2+3 x) \]

[Out]

-1/128/x^2+9/128/x+9/128/(2+3*x)^2+27/128/(2+3*x)+27/128*ln(x)-27/128*ln(2+3*x)

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Rubi [A]
time = 0.01, antiderivative size = 53, 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}{128 x^2}+\frac {9}{128 x}+\frac {27}{128 (3 x+2)}+\frac {9}{128 (3 x+2)^2}+\frac {27 \log (x)}{128}-\frac {27}{128} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(4 + 6*x)^3),x]

[Out]

-1/128*1/x^2 + 9/(128*x) + 9/(128*(2 + 3*x)^2) + 27/(128*(2 + 3*x)) + (27*Log[x])/128 - (27*Log[2 + 3*x])/128

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^3 (4+6 x)^3} \, dx &=\int \left (\frac {1}{64 x^3}-\frac {9}{128 x^2}+\frac {27}{128 x}-\frac {27}{64 (2+3 x)^3}-\frac {81}{128 (2+3 x)^2}-\frac {81}{128 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{128 x^2}+\frac {9}{128 x}+\frac {9}{128 (2+3 x)^2}+\frac {27}{128 (2+3 x)}+\frac {27 \log (x)}{128}-\frac {27}{128} \log (2+3 x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 44, normalized size = 0.83 \begin {gather*} \frac {1}{128} \left (\frac {2 \left (-2+12 x+81 x^2+81 x^3\right )}{x^2 (2+3 x)^2}+27 \log (x)-27 \log (2+3 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(4 + 6*x)^3),x]

[Out]

((2*(-2 + 12*x + 81*x^2 + 81*x^3))/(x^2*(2 + 3*x)^2) + 27*Log[x] - 27*Log[2 + 3*x])/128

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Mathics [A]
time = 1.91, size = 56, normalized size = 1.06 \begin {gather*} \frac {-4+24 x+27 x^2 \left (4+12 x+9 x^2\right ) \left (\text {Log}\left [x\right ]-\text {Log}\left [\frac {2}{3}+x\right ]\right )+162 x^2+162 x^3}{128 x^2 \left (4+12 x+9 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(x^3*(4 + 6*x)^3),x]')

[Out]

(-4 + 24 x + 27 x ^ 2 (4 + 12 x + 9 x ^ 2) (Log[x] - Log[2 / 3 + x]) + 162 x ^ 2 + 162 x ^ 3) / (128 x ^ 2 (4
+ 12 x + 9 x ^ 2))

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Maple [A]
time = 0.10, size = 42, normalized size = 0.79

method	result	size
norman	\(\frac {-\frac {1}{32}-\frac {81}{32} x^{3}-\frac {729}{256} x^{4}+\frac {3}{16} x}{x^{2} \left (2+3 x \right )^{2}}+\frac {27 \ln \left (x \right )}{128}-\frac {27 \ln \left (2+3 x \right )}{128}\)	\(40\)
risch	\(\frac {\frac {81}{64} x^{3}+\frac {81}{64} x^{2}+\frac {3}{16} x -\frac {1}{32}}{x^{2} \left (2+3 x \right )^{2}}+\frac {27 \ln \left (x \right )}{128}-\frac {27 \ln \left (2+3 x \right )}{128}\)	\(41\)
default	\(-\frac {1}{128 x^{2}}+\frac {9}{128 x}+\frac {9}{128 \left (2+3 x \right )^{2}}+\frac {27}{128 \left (2+3 x \right )}+\frac {27 \ln \left (x \right )}{128}-\frac {27 \ln \left (2+3 x \right )}{128}\)	\(42\)
meijerg	\(-\frac {1}{128 x^{2}}+\frac {9}{128 x}+\frac {63}{512}+\frac {27 \ln \left (x \right )}{128}-\frac {27 \ln \left (2\right )}{128}+\frac {27 \ln \left (3\right )}{128}-\frac {27 x \left (\frac {21 x}{2}+8\right )}{1024 \left (1+\frac {3 x}{2}\right )^{2}}-\frac {27 \ln \left (1+\frac {3 x}{2}\right )}{128}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(4+6*x)^3,x,method=_RETURNVERBOSE)

[Out]

-1/128/x^2+9/128/x+9/128/(2+3*x)^2+27/128/(2+3*x)+27/128*ln(x)-27/128*ln(2+3*x)

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Maxima [A]
time = 0.25, size = 48, normalized size = 0.91 \begin {gather*} \frac {81 \, x^{3} + 81 \, x^{2} + 12 \, x - 2}{64 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )}} - \frac {27}{128} \, \log \left (3 \, x + 2\right ) + \frac {27}{128} \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(4+6*x)^3,x, algorithm="maxima")

[Out]

1/64*(81*x^3 + 81*x^2 + 12*x - 2)/(9*x^4 + 12*x^3 + 4*x^2) - 27/128*log(3*x + 2) + 27/128*log(x)

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Fricas [A]
time = 0.31, size = 79, normalized size = 1.49 \begin {gather*} \frac {162 \, x^{3} + 162 \, x^{2} - 27 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )} \log \left (3 \, x + 2\right ) + 27 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )} \log \left (x\right ) + 24 \, x - 4}{128 \, {\left (9 \, x^{4} + 12 \, x^{3} + 4 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(4+6*x)^3,x, algorithm="fricas")

[Out]

1/128*(162*x^3 + 162*x^2 - 27*(9*x^4 + 12*x^3 + 4*x^2)*log(3*x + 2) + 27*(9*x^4 + 12*x^3 + 4*x^2)*log(x) + 24*
x - 4)/(9*x^4 + 12*x^3 + 4*x^2)

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Sympy [A]
time = 0.08, size = 46, normalized size = 0.87 \begin {gather*} \frac {27 \log {\left (x \right )}}{128} - \frac {27 \log {\left (x + \frac {2}{3} \right )}}{128} + \frac {81 x^{3} + 81 x^{2} + 12 x - 2}{576 x^{4} + 768 x^{3} + 256 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(4+6*x)**3,x)

[Out]

27*log(x)/128 - 27*log(x + 2/3)/128 + (81*x**3 + 81*x**2 + 12*x - 2)/(576*x**4 + 768*x**3 + 256*x**2)

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Giac [A]
time = 0.00, size = 53, normalized size = 1.00 \begin {gather*} \frac {27}{128} \ln \left |x\right |-\frac {27}{128} \ln \left |3 x+2\right |-\frac {-81 x^{3}-81 x^{2}-12 x+2}{64 \left (3 x^{2}+2 x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(4+6*x)^3,x)

[Out]

1/64*(81*x^3 + 81*x^2 + 12*x - 2)/(3*x^2 + 2*x)^2 - 27/128*log(abs(3*x + 2)) + 27/128*log(abs(x))

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Mupad [B]
time = 0.09, size = 41, normalized size = 0.77 \begin {gather*} \frac {\frac {9\,x^3}{64}+\frac {9\,x^2}{64}+\frac {x}{48}-\frac {1}{288}}{x^4+\frac {4\,x^3}{3}+\frac {4\,x^2}{9}}-\frac {27\,\mathrm {atanh}\left (3\,x+1\right )}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(6*x + 4)^3),x)

[Out]

(x/48 + (9*x^2)/64 + (9*x^3)/64 - 1/288)/((4*x^2)/9 + (4*x^3)/3 + x^4) - (27*atanh(3*x + 1))/64

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