3.269 \(\int \frac{1}{x^4 (4+6 x)^3} \, dx\)

Optimal. Leaf size=60 \[ \frac{9}{256 x^2}-\frac{1}{192 x^3}-\frac{27}{128 x}-\frac{27}{64 (3 x+2)}-\frac{27}{256 (3 x+2)^2}-\frac{135 \log (x)}{256}+\frac{135}{256} \log (3 x+2) \]

[Out]

-1/(192*x^3) + 9/(256*x^2) - 27/(128*x) - 27/(256*(2 + 3*x)^2) - 27/(64*(2 + 3*x)) - (135*Log[x])/256 + (135*L
og[2 + 3*x])/256

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Rubi [A]  time = 0.0207767, antiderivative size = 60, 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} \[ \frac{9}{256 x^2}-\frac{1}{192 x^3}-\frac{27}{128 x}-\frac{27}{64 (3 x+2)}-\frac{27}{256 (3 x+2)^2}-\frac{135 \log (x)}{256}+\frac{135}{256} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^4*(4 + 6*x)^3),x]

[Out]

-1/(192*x^3) + 9/(256*x^2) - 27/(128*x) - 27/(256*(2 + 3*x)^2) - 27/(64*(2 + 3*x)) - (135*Log[x])/256 + (135*L
og[2 + 3*x])/256

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)^3} \, dx &=\int \left (\frac{1}{64 x^4}-\frac{9}{128 x^3}+\frac{27}{128 x^2}-\frac{135}{256 x}+\frac{81}{128 (2+3 x)^3}+\frac{81}{64 (2+3 x)^2}+\frac{405}{256 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{192 x^3}+\frac{9}{256 x^2}-\frac{27}{128 x}-\frac{27}{256 (2+3 x)^2}-\frac{27}{64 (2+3 x)}-\frac{135 \log (x)}{256}+\frac{135}{256} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0320831, size = 49, normalized size = 0.82 \[ \frac{1}{768} \left (-\frac{2 \left (1215 x^4+1215 x^3+180 x^2-30 x+8\right )}{x^3 (3 x+2)^2}-405 \log (x)+405 \log (3 x+2)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^4*(4 + 6*x)^3),x]

[Out]

((-2*(8 - 30*x + 180*x^2 + 1215*x^3 + 1215*x^4))/(x^3*(2 + 3*x)^2) - 405*Log[x] + 405*Log[2 + 3*x])/768

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Maple [A]  time = 0.01, size = 47, normalized size = 0.8 \begin{align*} -{\frac{1}{192\,{x}^{3}}}+{\frac{9}{256\,{x}^{2}}}-{\frac{27}{128\,x}}-{\frac{27}{256\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{27}{128+192\,x}}-{\frac{135\,\ln \left ( x \right ) }{256}}+{\frac{135\,\ln \left ( 2+3\,x \right ) }{256}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192/x^3+9/256/x^2-27/128/x-27/256/(2+3*x)^2-27/64/(2+3*x)-135/256*ln(x)+135/256*ln(2+3*x)

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Maxima [A]  time = 1.00904, size = 72, normalized size = 1.2 \begin{align*} -\frac{1215 \, x^{4} + 1215 \, x^{3} + 180 \, x^{2} - 30 \, x + 8}{384 \,{\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )}} + \frac{135}{256} \, \log \left (3 \, x + 2\right ) - \frac{135}{256} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(1215*x^4 + 1215*x^3 + 180*x^2 - 30*x + 8)/(9*x^5 + 12*x^4 + 4*x^3) + 135/256*log(3*x + 2) - 135/256*lo
g(x)

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Fricas [A]  time = 1.79541, size = 215, normalized size = 3.58 \begin{align*} -\frac{2430 \, x^{4} + 2430 \, x^{3} + 360 \, x^{2} - 405 \,{\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )} \log \left (3 \, x + 2\right ) + 405 \,{\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )} \log \left (x\right ) - 60 \, x + 16}{768 \,{\left (9 \, x^{5} + 12 \, x^{4} + 4 \, x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/768*(2430*x^4 + 2430*x^3 + 360*x^2 - 405*(9*x^5 + 12*x^4 + 4*x^3)*log(3*x + 2) + 405*(9*x^5 + 12*x^4 + 4*x^
3)*log(x) - 60*x + 16)/(9*x^5 + 12*x^4 + 4*x^3)

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Sympy [A]  time = 0.204437, size = 51, normalized size = 0.85 \begin{align*} - \frac{135 \log{\left (x \right )}}{256} + \frac{135 \log{\left (x + \frac{2}{3} \right )}}{256} - \frac{1215 x^{4} + 1215 x^{3} + 180 x^{2} - 30 x + 8}{3456 x^{5} + 4608 x^{4} + 1536 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-135*log(x)/256 + 135*log(x + 2/3)/256 - (1215*x**4 + 1215*x**3 + 180*x**2 - 30*x + 8)/(3456*x**5 + 4608*x**4
+ 1536*x**3)

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Giac [A]  time = 1.19775, size = 63, normalized size = 1.05 \begin{align*} -\frac{1215 \, x^{4} + 1215 \, x^{3} + 180 \, x^{2} - 30 \, x + 8}{384 \,{\left (3 \, x + 2\right )}^{2} x^{3}} + \frac{135}{256} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{135}{256} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(1215*x^4 + 1215*x^3 + 180*x^2 - 30*x + 8)/((3*x + 2)^2*x^3) + 135/256*log(abs(3*x + 2)) - 135/256*log(
abs(x))