3.3.63 \(\int \frac {1}{x^3 (4+6 x)^2} \, dx\)

Optimal. Leaf size=42 \[ -\frac {1}{32 x^2}+\frac {3}{16 x}+\frac {9}{32 (3 x+2)}+\frac {27 \log (x)}{64}-\frac {27}{64} \log (3 x+2) \]

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Rubi [A]  time = 0.01, antiderivative size = 42, 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 = {44} \begin {gather*} -\frac {1}{32 x^2}+\frac {3}{16 x}+\frac {9}{32 (3 x+2)}+\frac {27 \log (x)}{64}-\frac {27}{64} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(4 + 6*x)^2),x]

[Out]

-1/(32*x^2) + 3/(16*x) + 9/(32*(2 + 3*x)) + (27*Log[x])/64 - (27*Log[2 + 3*x])/64

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^3 (4+6 x)^2} \, dx &=\int \left (\frac {1}{16 x^3}-\frac {3}{16 x^2}+\frac {27}{64 x}-\frac {27}{32 (2+3 x)^2}-\frac {81}{64 (2+3 x)}\right ) \, dx\\ &=-\frac {1}{32 x^2}+\frac {3}{16 x}+\frac {9}{32 (2+3 x)}+\frac {27 \log (x)}{64}-\frac {27}{64} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 36, normalized size = 0.86 \begin {gather*} \frac {1}{64} \left (-\frac {2}{x^2}+\frac {12}{x}+\frac {18}{3 x+2}+27 \log (x)-27 \log (3 x+2)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(4 + 6*x)^2),x]

[Out]

(-2/x^2 + 12/x + 18/(2 + 3*x) + 27*Log[x] - 27*Log[2 + 3*x])/64

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 (4+6 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/(x^3*(4 + 6*x)^2),x]

[Out]

IntegrateAlgebraic[1/(x^3*(4 + 6*x)^2), x]

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fricas [A]  time = 1.07, size = 59, normalized size = 1.40 \begin {gather*} \frac {54 \, x^{2} - 27 \, {\left (3 \, x^{3} + 2 \, x^{2}\right )} \log \left (3 \, x + 2\right ) + 27 \, {\left (3 \, x^{3} + 2 \, x^{2}\right )} \log \relax (x) + 18 \, x - 4}{64 \, {\left (3 \, x^{3} + 2 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(54*x^2 - 27*(3*x^3 + 2*x^2)*log(3*x + 2) + 27*(3*x^3 + 2*x^2)*log(x) + 18*x - 4)/(3*x^3 + 2*x^2)

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giac [A]  time = 0.94, size = 51, normalized size = 1.21 \begin {gather*} \frac {9}{32 \, {\left (3 \, x + 2\right )}} - \frac {9 \, {\left (\frac {12}{3 \, x + 2} - 5\right )}}{128 \, {\left (\frac {2}{3 \, x + 2} - 1\right )}^{2}} + \frac {27}{64} \, \log \left ({\left | -\frac {2}{3 \, x + 2} + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/32/(3*x + 2) - 9/128*(12/(3*x + 2) - 5)/(2/(3*x + 2) - 1)^2 + 27/64*log(abs(-2/(3*x + 2) + 1))

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maple [A]  time = 0.01, size = 33, normalized size = 0.79 \begin {gather*} \frac {27 \ln \relax (x )}{64}-\frac {27 \ln \left (3 x +2\right )}{64}+\frac {3}{16 x}-\frac {1}{32 x^{2}}+\frac {9}{32 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32/x^2+3/16/x+9/32/(3*x+2)+27/64*ln(x)-27/64*ln(3*x+2)

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maxima [A]  time = 1.31, size = 38, normalized size = 0.90 \begin {gather*} \frac {27 \, x^{2} + 9 \, x - 2}{32 \, {\left (3 \, x^{3} + 2 \, x^{2}\right )}} - \frac {27}{64} \, \log \left (3 \, x + 2\right ) + \frac {27}{64} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(27*x^2 + 9*x - 2)/(3*x^3 + 2*x^2) - 27/64*log(3*x + 2) + 27/64*log(x)

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mupad [B]  time = 0.04, size = 31, normalized size = 0.74 \begin {gather*} \frac {\frac {9\,x^2}{32}+\frac {3\,x}{32}-\frac {1}{48}}{x^3+\frac {2\,x^2}{3}}-\frac {27\,\mathrm {atanh}\left (3\,x+1\right )}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(6*x + 4)^2),x)

[Out]

((3*x)/32 + (9*x^2)/32 - 1/48)/((2*x^2)/3 + x^3) - (27*atanh(3*x + 1))/32

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sympy [A]  time = 0.16, size = 36, normalized size = 0.86 \begin {gather*} \frac {27 \log {\relax (x )}}{64} - \frac {27 \log {\left (x + \frac {2}{3} \right )}}{64} + \frac {27 x^{2} + 9 x - 2}{96 x^{3} + 64 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27*log(x)/64 - 27*log(x + 2/3)/64 + (27*x**2 + 9*x - 2)/(96*x**3 + 64*x**2)

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