3.264 \(\int \frac{1}{x^4 (4+6 x)^2} \, dx\)

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

[Out]

-1/(48*x^3) + 3/(32*x^2) - 27/(64*x) - 27/(64*(2 + 3*x)) - (27*Log[x])/32 + (27*Log[2 + 3*x])/32

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Rubi [A]  time = 0.0164746, antiderivative size = 49, 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} \[ \frac{3}{32 x^2}-\frac{1}{48 x^3}-\frac{27}{64 x}-\frac{27}{64 (3 x+2)}-\frac{27 \log (x)}{32}+\frac{27}{32} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(4 + 6*x)^2),x]

[Out]

-1/(48*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 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^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*}

Mathematica [A]  time = 0.0453362, size = 44, normalized size = 0.9 \[ \frac{1}{192} \left (-\frac{4 \left (81 x^3+27 x^2-6 x+2\right )}{x^3 (3 x+2)}-162 \log (x)+162 \log (3 x+2)\right ) \]

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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Maple [A]  time = 0.007, size = 38, normalized size = 0.8 \begin{align*} -{\frac{1}{48\,{x}^{3}}}+{\frac{3}{32\,{x}^{2}}}-{\frac{27}{64\,x}}-{\frac{27}{128+192\,x}}-{\frac{27\,\ln \left ( x \right ) }{32}}+{\frac{27\,\ln \left ( 2+3\,x \right ) }{32}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(4+6*x)^2,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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Maxima [A]  time = 1.0237, size = 58, normalized size = 1.18 \begin{align*} -\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{align*}

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 = 1.71794, size = 155, normalized size = 3.16 \begin{align*} -\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{align*}

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 = 0.164463, size = 41, normalized size = 0.84 \begin{align*} - \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{align*}

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 = 1.22874, size = 81, normalized size = 1.65 \begin{align*} -\frac{27}{64 \,{\left (3 \, x + 2\right )}} - \frac{9 \,{\left (\frac{60}{3 \, x + 2} - \frac{72}{{\left (3 \, x + 2\right )}^{2}} - 13\right )}}{128 \,{\left (\frac{2}{3 \, x + 2} - 1\right )}^{3}} - \frac{27}{32} \, \log \left ({\left | -\frac{2}{3 \, x + 2} + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/64/(3*x + 2) - 9/128*(60/(3*x + 2) - 72/(3*x + 2)^2 - 13)/(2/(3*x + 2) - 1)^3 - 27/32*log(abs(-2/(3*x + 2)
+ 1))