3.70.9 \(\int \frac {-864+3096 x+1438 x^2-1719 x^3-4 x^4+4 x^5}{432 x^2-1764 x^3+1873 x^4-148 x^5+4 x^6} \, dx\)

Optimal. Leaf size=24 \[ \frac {2}{x-2 x^2}+\log \left (12 \left (-3+\frac {36}{x}\right )+x\right ) \]

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Rubi [A]  time = 0.13, antiderivative size = 28, normalized size of antiderivative = 1.17, number of steps used = 3, number of rules used = 2, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2074, 628} \begin {gather*} \log \left (x^2-36 x+432\right )+\frac {4}{1-2 x}+\frac {2}{x}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-864 + 3096*x + 1438*x^2 - 1719*x^3 - 4*x^4 + 4*x^5)/(432*x^2 - 1764*x^3 + 1873*x^4 - 148*x^5 + 4*x^6),x]

[Out]

4/(1 - 2*x) + 2/x - Log[x] + Log[432 - 36*x + x^2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2}{x^2}-\frac {1}{x}+\frac {8}{(-1+2 x)^2}+\frac {2 (-18+x)}{432-36 x+x^2}\right ) \, dx\\ &=\frac {4}{1-2 x}+\frac {2}{x}-\log (x)+2 \int \frac {-18+x}{432-36 x+x^2} \, dx\\ &=\frac {4}{1-2 x}+\frac {2}{x}-\log (x)+\log \left (432-36 x+x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 25, normalized size = 1.04 \begin {gather*} \frac {2}{x-2 x^2}-\log (x)+\log \left (432-36 x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-864 + 3096*x + 1438*x^2 - 1719*x^3 - 4*x^4 + 4*x^5)/(432*x^2 - 1764*x^3 + 1873*x^4 - 148*x^5 + 4*x
^6),x]

[Out]

2/(x - 2*x^2) - Log[x] + Log[432 - 36*x + x^2]

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fricas [A]  time = 0.99, size = 46, normalized size = 1.92 \begin {gather*} \frac {{\left (2 \, x^{2} - x\right )} \log \left (x^{2} - 36 \, x + 432\right ) - {\left (2 \, x^{2} - x\right )} \log \relax (x) - 2}{2 \, x^{2} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^5-4*x^4-1719*x^3+1438*x^2+3096*x-864)/(4*x^6-148*x^5+1873*x^4-1764*x^3+432*x^2),x, algorithm="f
ricas")

[Out]

((2*x^2 - x)*log(x^2 - 36*x + 432) - (2*x^2 - x)*log(x) - 2)/(2*x^2 - x)

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giac [A]  time = 0.20, size = 27, normalized size = 1.12 \begin {gather*} -\frac {2}{{\left (2 \, x - 1\right )} x} + \log \left (x^{2} - 36 \, x + 432\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^5-4*x^4-1719*x^3+1438*x^2+3096*x-864)/(4*x^6-148*x^5+1873*x^4-1764*x^3+432*x^2),x, algorithm="g
iac")

[Out]

-2/((2*x - 1)*x) + log(x^2 - 36*x + 432) - log(abs(x))

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maple [A]  time = 0.04, size = 27, normalized size = 1.12

method	result	size
norman	\(-\frac {2}{x \left (2 x -1\right )}-\ln \relax (x )+\ln \left (x^{2}-36 x +432\right )\)	\(27\)
risch	\(-\frac {2}{x \left (2 x -1\right )}-\ln \relax (x )+\ln \left (x^{2}-36 x +432\right )\)	\(27\)
default	\(-\frac {4}{2 x -1}+\frac {2}{x}-\ln \relax (x )+\ln \left (x^{2}-36 x +432\right )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^5-4*x^4-1719*x^3+1438*x^2+3096*x-864)/(4*x^6-148*x^5+1873*x^4-1764*x^3+432*x^2),x,method=_RETURNVERBO
SE)

[Out]

-2/x/(2*x-1)-ln(x)+ln(x^2-36*x+432)

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maxima [A]  time = 0.36, size = 27, normalized size = 1.12 \begin {gather*} -\frac {2}{2 \, x^{2} - x} + \log \left (x^{2} - 36 \, x + 432\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^5-4*x^4-1719*x^3+1438*x^2+3096*x-864)/(4*x^6-148*x^5+1873*x^4-1764*x^3+432*x^2),x, algorithm="m
axima")

[Out]

-2/(2*x^2 - x) + log(x^2 - 36*x + 432) - log(x)

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mupad [B]  time = 0.09, size = 25, normalized size = 1.04 \begin {gather*} \ln \left (x^2-36\,x+432\right )-\ln \relax (x)+\frac {1}{\frac {x}{2}-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3096*x + 1438*x^2 - 1719*x^3 - 4*x^4 + 4*x^5 - 864)/(432*x^2 - 1764*x^3 + 1873*x^4 - 148*x^5 + 4*x^6),x)

[Out]

log(x^2 - 36*x + 432) - log(x) + 1/(x/2 - x^2)

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sympy [A]  time = 0.14, size = 20, normalized size = 0.83 \begin {gather*} - \log {\relax (x )} + \log {\left (x^{2} - 36 x + 432 \right )} - \frac {2}{2 x^{2} - x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**5-4*x**4-1719*x**3+1438*x**2+3096*x-864)/(4*x**6-148*x**5+1873*x**4-1764*x**3+432*x**2),x)

[Out]

-log(x) + log(x**2 - 36*x + 432) - 2/(2*x**2 - x)

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