3.64.89 \(\int \frac {-8+6 x+181 x^2-90 x^3}{-8 x+4 x^2} \, dx\)

Optimal. Leaf size=21 \[ \frac {x}{4}-\frac {45 x^2}{4}+\log (-2+x)+\log (2 x) \]

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Rubi [A]  time = 0.04, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1593, 1620} \begin {gather*} -\frac {45 x^2}{4}+\frac {x}{4}+\log (2-x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8 + 6*x + 181*x^2 - 90*x^3)/(-8*x + 4*x^2),x]

[Out]

x/4 - (45*x^2)/4 + Log[2 - x] + Log[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 {-8+6 x+181 x^2-90 x^3}{x (-8+4 x)} \, dx\\ &=\int \left (\frac {1}{4}+\frac {1}{-2+x}+\frac {1}{x}-\frac {45 x}{2}\right ) \, dx\\ &=\frac {x}{4}-\frac {45 x^2}{4}+\log (2-x)+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} \frac {x}{4}-\frac {45 x^2}{4}+\log (2-x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 + 6*x + 181*x^2 - 90*x^3)/(-8*x + 4*x^2),x]

[Out]

x/4 - (45*x^2)/4 + Log[2 - x] + Log[x]

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fricas [A]  time = 0.64, size = 17, normalized size = 0.81 \begin {gather*} -\frac {45}{4} \, x^{2} + \frac {1}{4} \, x + \log \left (x^{2} - 2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-90*x^3+181*x^2+6*x-8)/(4*x^2-8*x),x, algorithm="fricas")

[Out]

-45/4*x^2 + 1/4*x + log(x^2 - 2*x)

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giac [A]  time = 0.14, size = 17, normalized size = 0.81 \begin {gather*} -\frac {45}{4} \, x^{2} + \frac {1}{4} \, x + \log \left ({\left | x - 2 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-90*x^3+181*x^2+6*x-8)/(4*x^2-8*x),x, algorithm="giac")

[Out]

-45/4*x^2 + 1/4*x + log(abs(x - 2)) + log(abs(x))

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maple [A]  time = 0.41, size = 16, normalized size = 0.76

method	result	size
default	\(-\frac {45 x^{2}}{4}+\frac {x}{4}+\ln \relax (x )+\ln \left (x -2\right )\)	\(16\)
norman	\(-\frac {45 x^{2}}{4}+\frac {x}{4}+\ln \relax (x )+\ln \left (x -2\right )\)	\(16\)
risch	\(\frac {x}{4}-\frac {45 x^{2}}{4}+\ln \left (x^{2}-2 x \right )\)	\(18\)
meijerg	\(\ln \left (1-\frac {x}{2}\right )+\ln \relax (x )-\ln \relax (2)+i \pi -\frac {15 x \left (\frac {3 x}{2}+6\right )}{2}+\frac {181 x}{4}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-90*x^3+181*x^2+6*x-8)/(4*x^2-8*x),x,method=_RETURNVERBOSE)

[Out]

-45/4*x^2+1/4*x+ln(x)+ln(x-2)

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maxima [A]  time = 0.49, size = 15, normalized size = 0.71 \begin {gather*} -\frac {45}{4} \, x^{2} + \frac {1}{4} \, x + \log \left (x - 2\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-90*x^3+181*x^2+6*x-8)/(4*x^2-8*x),x, algorithm="maxima")

[Out]

-45/4*x^2 + 1/4*x + log(x - 2) + log(x)

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mupad [B]  time = 4.08, size = 15, normalized size = 0.71 \begin {gather*} \frac {x}{4}+\ln \left (x\,\left (x-2\right )\right )-\frac {45\,x^2}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x + 181*x^2 - 90*x^3 - 8)/(8*x - 4*x^2),x)

[Out]

x/4 + log(x*(x - 2)) - (45*x^2)/4

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sympy [A]  time = 0.08, size = 17, normalized size = 0.81 \begin {gather*} - \frac {45 x^{2}}{4} + \frac {x}{4} + \log {\left (x^{2} - 2 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-90*x**3+181*x**2+6*x-8)/(4*x**2-8*x),x)

[Out]

-45*x**2/4 + x/4 + log(x**2 - 2*x)

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