3.62.19 \(\int \frac {-100-12 x+x^3}{20 x+4 x^2+x^4} \, dx\)

Optimal. Leaf size=20 \[ \log \left (\frac {\left (x+\frac {4 (5+x)}{x^2}\right )^2}{5 x}\right ) \]

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Rubi [A]  time = 0.11, antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1594, 6742, 1587} \begin {gather*} 2 \log \left (x^3+4 x+20\right )-5 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-100 - 12*x + x^3)/(20*x + 4*x^2 + x^4),x]

[Out]

-5*Log[x] + 2*Log[20 + 4*x + x^3]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 1594

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-100-12 x+x^3}{x \left (20+4 x+x^3\right )} \, dx\\ &=\int \left (-\frac {5}{x}+\frac {2 \left (4+3 x^2\right )}{20+4 x+x^3}\right ) \, dx\\ &=-5 \log (x)+2 \int \frac {4+3 x^2}{20+4 x+x^3} \, dx\\ &=-5 \log (x)+2 \log \left (20+4 x+x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.80 \begin {gather*} -5 \log (x)+2 \log \left (20+4 x+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-100 - 12*x + x^3)/(20*x + 4*x^2 + x^4),x]

[Out]

-5*Log[x] + 2*Log[20 + 4*x + x^3]

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fricas [A]  time = 0.53, size = 16, normalized size = 0.80 \begin {gather*} 2 \, \log \left (x^{3} + 4 \, x + 20\right ) - 5 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-12*x-100)/(x^4+4*x^2+20*x),x, algorithm="fricas")

[Out]

2*log(x^3 + 4*x + 20) - 5*log(x)

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giac [A]  time = 0.13, size = 18, normalized size = 0.90 \begin {gather*} 2 \, \log \left ({\left | x^{3} + 4 \, x + 20 \right |}\right ) - 5 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-12*x-100)/(x^4+4*x^2+20*x),x, algorithm="giac")

[Out]

2*log(abs(x^3 + 4*x + 20)) - 5*log(abs(x))

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maple [A]  time = 0.02, size = 17, normalized size = 0.85

method	result	size
default	\(-5 \ln \relax (x )+2 \ln \left (x^{3}+4 x +20\right )\)	\(17\)
norman	\(-5 \ln \relax (x )+2 \ln \left (x^{3}+4 x +20\right )\)	\(17\)
risch	\(-5 \ln \relax (x )+2 \ln \left (x^{3}+4 x +20\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-12*x-100)/(x^4+4*x^2+20*x),x,method=_RETURNVERBOSE)

[Out]

-5*ln(x)+2*ln(x^3+4*x+20)

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maxima [A]  time = 0.42, size = 16, normalized size = 0.80 \begin {gather*} 2 \, \log \left (x^{3} + 4 \, x + 20\right ) - 5 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-12*x-100)/(x^4+4*x^2+20*x),x, algorithm="maxima")

[Out]

2*log(x^3 + 4*x + 20) - 5*log(x)

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mupad [B]  time = 0.07, size = 16, normalized size = 0.80 \begin {gather*} 2\,\ln \left (x^3+4\,x+20\right )-5\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12*x - x^3 + 100)/(20*x + 4*x^2 + x^4),x)

[Out]

2*log(4*x + x^3 + 20) - 5*log(x)

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sympy [A]  time = 0.10, size = 15, normalized size = 0.75 \begin {gather*} - 5 \log {\relax (x )} + 2 \log {\left (x^{3} + 4 x + 20 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-12*x-100)/(x**4+4*x**2+20*x),x)

[Out]

-5*log(x) + 2*log(x**3 + 4*x + 20)

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