3.3.67 \(\int \frac {1}{x^2 (4+6 x)^3} \, dx\) [267]

Optimal. Leaf size=46 \[ -\frac {1}{64 x}-\frac {3}{64 (2+3 x)^2}-\frac {3}{32 (2+3 x)}-\frac {9 \log (x)}{128}+\frac {9}{128} \log (2+3 x) \]

[Out]

-1/64/x-3/64/(2+3*x)^2-3/32/(2+3*x)-9/128*ln(x)+9/128*ln(2+3*x)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 46, 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}{64 x}-\frac {3}{32 (3 x+2)}-\frac {3}{64 (3 x+2)^2}-\frac {9 \log (x)}{128}+\frac {9}{128} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(4 + 6*x)^3),x]

[Out]

-1/64*1/x - 3/(64*(2 + 3*x)^2) - 3/(32*(2 + 3*x)) - (9*Log[x])/128 + (9*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^2 (4+6 x)^3} \, dx &=\int \left (\frac {1}{64 x^2}-\frac {9}{128 x}+\frac {9}{32 (2+3 x)^3}+\frac {9}{32 (2+3 x)^2}+\frac {27}{128 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{64 x}-\frac {3}{64 (2+3 x)^2}-\frac {3}{32 (2+3 x)}-\frac {9 \log (x)}{128}+\frac {9}{128} \log (2+3 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 39, normalized size = 0.85 \begin {gather*} \frac {1}{128} \left (-\frac {2 \left (4+27 x+27 x^2\right )}{x (2+3 x)^2}-9 \log (x)+9 \log (2+3 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(4 + 6*x)^3),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 37, normalized size = 0.80

method	result	size
risch	\(\frac {-\frac {27}{64} x^{2}-\frac {27}{64} x -\frac {1}{16}}{x \left (2+3 x \right )^{2}}-\frac {9 \ln \left (x \right )}{128}+\frac {9 \ln \left (2+3 x \right )}{128}\)	\(36\)
default	\(-\frac {1}{64 x}-\frac {3}{64 \left (2+3 x \right )^{2}}-\frac {3}{32 \left (2+3 x \right )}-\frac {9 \ln \left (x \right )}{128}+\frac {9 \ln \left (2+3 x \right )}{128}\)	\(37\)
norman	\(\frac {-\frac {1}{16}+\frac {27}{32} x^{2}+\frac {243}{256} x^{3}}{x \left (2+3 x \right )^{2}}-\frac {9 \ln \left (x \right )}{128}+\frac {9 \ln \left (2+3 x \right )}{128}\)	\(37\)
meijerg	\(-\frac {1}{64 x}-\frac {15}{256}-\frac {9 \ln \left (x \right )}{128}+\frac {9 \ln \left (2\right )}{128}-\frac {9 \ln \left (3\right )}{128}+\frac {9 x \left (\frac {15 x}{2}+6\right )}{512 \left (1+\frac {3 x}{2}\right )^{2}}+\frac {9 \ln \left (1+\frac {3 x}{2}\right )}{128}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(4+6*x)^3,x,method=_RETURNVERBOSE)

[Out]

-1/64/x-3/64/(2+3*x)^2-3/32/(2+3*x)-9/128*ln(x)+9/128*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 41, normalized size = 0.89 \begin {gather*} -\frac {27 \, x^{2} + 27 \, x + 4}{64 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )}} + \frac {9}{128} \, \log \left (3 \, x + 2\right ) - \frac {9}{128} \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(4+6*x)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.67, size = 68, normalized size = 1.48 \begin {gather*} -\frac {54 \, x^{2} - 9 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )} \log \left (3 \, x + 2\right ) + 9 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )} \log \left (x\right ) + 54 \, x + 8}{128 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(4+6*x)^3,x, algorithm="fricas")

[Out]

-1/128*(54*x^2 - 9*(9*x^3 + 12*x^2 + 4*x)*log(3*x + 2) + 9*(9*x^3 + 12*x^2 + 4*x)*log(x) + 54*x + 8)/(9*x^3 +
12*x^2 + 4*x)

________________________________________________________________________________________

Sympy [A]
time = 0.05, size = 41, normalized size = 0.89 \begin {gather*} \frac {- 27 x^{2} - 27 x - 4}{576 x^{3} + 768 x^{2} + 256 x} - \frac {9 \log {\left (x \right )}}{128} + \frac {9 \log {\left (x + \frac {2}{3} \right )}}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(4+6*x)**3,x)

[Out]

(-27*x**2 - 27*x - 4)/(576*x**3 + 768*x**2 + 256*x) - 9*log(x)/128 + 9*log(x + 2/3)/128

________________________________________________________________________________________

Giac [A]
time = 0.83, size = 37, normalized size = 0.80 \begin {gather*} -\frac {27 \, x^{2} + 27 \, x + 4}{64 \, {\left (3 \, x + 2\right )}^{2} x} + \frac {9}{128} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {9}{128} \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(4+6*x)^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.09, size = 35, normalized size = 0.76 \begin {gather*} \frac {9\,\mathrm {atanh}\left (3\,x+1\right )}{64}-\frac {\frac {3\,x^2}{64}+\frac {3\,x}{64}+\frac {1}{144}}{x^3+\frac {4\,x^2}{3}+\frac {4\,x}{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(6*x + 4)^3),x)

[Out]

(9*atanh(3*x + 1))/64 - ((3*x)/64 + (3*x^2)/64 + 1/144)/((4*x)/9 + (4*x^2)/3 + x^3)

________________________________________________________________________________________