3.45.34 \(\int \frac {798-479 x+80 x^2-4 x^3}{399 x-80 x^2+4 x^3} \, dx\)

Optimal. Leaf size=23 \[ \log \left (\frac {e^{-x} x^2}{1-(-20+2 x)^2}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.04, 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 = {1594, 1628} \begin {gather*} -x-\log (19-2 x)-\log (21-2 x)+2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(798 - 479*x + 80*x^2 - 4*x^3)/(399*x - 80*x^2 + 4*x^3),x]

[Out]

-x - Log[19 - 2*x] - Log[21 - 2*x] + 2*Log[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 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {798-479 x+80 x^2-4 x^3}{x \left (399-80 x+4 x^2\right )} \, dx\\ &=\int \left (-1+\frac {2}{x}-\frac {2}{-21+2 x}-\frac {2}{-19+2 x}\right ) \, dx\\ &=-x-\log (19-2 x)-\log (21-2 x)+2 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.91 \begin {gather*} -x+2 \log (x)-\log \left (399-80 x+4 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(798 - 479*x + 80*x^2 - 4*x^3)/(399*x - 80*x^2 + 4*x^3),x]

[Out]

-x + 2*Log[x] - Log[399 - 80*x + 4*x^2]

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fricas [A]  time = 0.75, size = 21, normalized size = 0.91 \begin {gather*} -x - \log \left (4 \, x^{2} - 80 \, x + 399\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^3+80*x^2-479*x+798)/(4*x^3-80*x^2+399*x),x, algorithm="fricas")

[Out]

-x - log(4*x^2 - 80*x + 399) + 2*log(x)

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giac [A]  time = 0.16, size = 27, normalized size = 1.17 \begin {gather*} -x - \log \left ({\left | 2 \, x - 19 \right |}\right ) - \log \left ({\left | 2 \, x - 21 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^3+80*x^2-479*x+798)/(4*x^3-80*x^2+399*x),x, algorithm="giac")

[Out]

-x - log(abs(2*x - 19)) - log(abs(2*x - 21)) + 2*log(abs(x))

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maple [A]  time = 0.03, size = 22, normalized size = 0.96

method	result	size
risch	\(-x +2 \ln \relax (x )-\ln \left (4 x^{2}-80 x +399\right )\)	\(22\)
default	\(-x -\ln \left (2 x -21\right )+2 \ln \relax (x )-\ln \left (2 x -19\right )\)	\(25\)
norman	\(-x -\ln \left (2 x -21\right )+2 \ln \relax (x )-\ln \left (2 x -19\right )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^3+80*x^2-479*x+798)/(4*x^3-80*x^2+399*x),x,method=_RETURNVERBOSE)

[Out]

-x+2*ln(x)-ln(4*x^2-80*x+399)

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maxima [A]  time = 0.35, size = 24, normalized size = 1.04 \begin {gather*} -x - \log \left (2 \, x - 19\right ) - \log \left (2 \, x - 21\right ) + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^3+80*x^2-479*x+798)/(4*x^3-80*x^2+399*x),x, algorithm="maxima")

[Out]

-x - log(2*x - 19) - log(2*x - 21) + 2*log(x)

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mupad [B]  time = 0.08, size = 19, normalized size = 0.83 \begin {gather*} 2\,\ln \relax (x)-\ln \left (x^2-20\,x+\frac {399}{4}\right )-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(479*x - 80*x^2 + 4*x^3 - 798)/(399*x - 80*x^2 + 4*x^3),x)

[Out]

2*log(x) - log(x^2 - 20*x + 399/4) - x

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sympy [A]  time = 0.10, size = 17, normalized size = 0.74 \begin {gather*} - x + 2 \log {\relax (x )} - \log {\left (4 x^{2} - 80 x + 399 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**3+80*x**2-479*x+798)/(4*x**3-80*x**2+399*x),x)

[Out]

-x + 2*log(x) - log(4*x**2 - 80*x + 399)

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