3.12.3 \(\int \frac {-4-48 x-2 x^2-13 x^3-12 x^4}{x^3+12 x^4} \, dx\)

Optimal. Leaf size=31 \[ -3+\frac {2}{x^2}-2 x-\log \left (\frac {e^{-x} x}{4+\frac {1}{3 x}}\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 19, normalized size of antiderivative = 0.61, number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1593, 1620} \begin {gather*} \frac {2}{x^2}-x-2 \log (x)+\log (12 x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 - 48*x - 2*x^2 - 13*x^3 - 12*x^4)/(x^3 + 12*x^4),x]

[Out]

2/x^2 - x - 2*Log[x] + Log[1 + 12*x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4-48 x-2 x^2-13 x^3-12 x^4}{x^3 (1+12 x)} \, dx\\ &=\int \left (-1-\frac {4}{x^3}-\frac {2}{x}+\frac {12}{1+12 x}\right ) \, dx\\ &=\frac {2}{x^2}-x-2 \log (x)+\log (1+12 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 0.61 \begin {gather*} \frac {2}{x^2}-x-2 \log (x)+\log (1+12 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 - 48*x - 2*x^2 - 13*x^3 - 12*x^4)/(x^3 + 12*x^4),x]

[Out]

2/x^2 - x - 2*Log[x] + Log[1 + 12*x]

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fricas [A]  time = 0.58, size = 28, normalized size = 0.90 \begin {gather*} -\frac {x^{3} - x^{2} \log \left (12 \, x + 1\right ) + 2 \, x^{2} \log \relax (x) - 2}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x^4-13*x^3-2*x^2-48*x-4)/(12*x^4+x^3),x, algorithm="fricas")

[Out]

-(x^3 - x^2*log(12*x + 1) + 2*x^2*log(x) - 2)/x^2

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giac [A]  time = 0.31, size = 21, normalized size = 0.68 \begin {gather*} -x + \frac {2}{x^{2}} + \log \left ({\left | 12 \, x + 1 \right |}\right ) - 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x^4-13*x^3-2*x^2-48*x-4)/(12*x^4+x^3),x, algorithm="giac")

[Out]

-x + 2/x^2 + log(abs(12*x + 1)) - 2*log(abs(x))

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maple [A]  time = 0.42, size = 20, normalized size = 0.65

method	result	size
default	\(-x +\frac {2}{x^{2}}-2 \ln \relax (x )+\ln \left (12 x +1\right )\)	\(20\)
risch	\(-x +\frac {2}{x^{2}}-2 \ln \relax (x )+\ln \left (12 x +1\right )\)	\(20\)
norman	\(\frac {-x^{3}+2}{x^{2}}-2 \ln \relax (x )+\ln \left (12 x +1\right )\)	\(23\)
meijerg	\(-x +\ln \left (12 x +1\right )-2 \ln \relax (x )-4 \ln \relax (2)-2 \ln \relax (3)+\frac {2}{x^{2}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-12*x^4-13*x^3-2*x^2-48*x-4)/(12*x^4+x^3),x,method=_RETURNVERBOSE)

[Out]

-x+2/x^2-2*ln(x)+ln(12*x+1)

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maxima [A]  time = 0.48, size = 19, normalized size = 0.61 \begin {gather*} -x + \frac {2}{x^{2}} + \log \left (12 \, x + 1\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x^4-13*x^3-2*x^2-48*x-4)/(12*x^4+x^3),x, algorithm="maxima")

[Out]

-x + 2/x^2 + log(12*x + 1) - 2*log(x)

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mupad [B]  time = 0.72, size = 17, normalized size = 0.55 \begin {gather*} \ln \left (x+\frac {1}{12}\right )-x-2\,\ln \relax (x)+\frac {2}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(48*x + 2*x^2 + 13*x^3 + 12*x^4 + 4)/(x^3 + 12*x^4),x)

[Out]

log(x + 1/12) - x - 2*log(x) + 2/x^2

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sympy [A]  time = 0.11, size = 17, normalized size = 0.55 \begin {gather*} - x - 2 \log {\relax (x )} + \log {\left (x + \frac {1}{12} \right )} + \frac {2}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x**4-13*x**3-2*x**2-48*x-4)/(12*x**4+x**3),x)

[Out]

-x - 2*log(x) + log(x + 1/12) + 2/x**2

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