3.158 \(\int \frac{1-4 x^2+x^3}{(-2+x)^4} \, dx\)

Optimal. Leaf size=36 \[ \frac{2}{2-x}+\frac{2}{(2-x)^2}-\frac{7}{3 (2-x)^3}+\log (2-x) \]

[Out]

-7/(3*(2 - x)^3) + 2/(2 - x)^2 + 2/(2 - x) + Log[2 - x]

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Rubi [A]  time = 0.0219846, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {1850} \[ \frac{2}{2-x}+\frac{2}{(2-x)^2}-\frac{7}{3 (2-x)^3}+\log (2-x) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 4*x^2 + x^3)/(-2 + x)^4,x]

[Out]

-7/(3*(2 - x)^3) + 2/(2 - x)^2 + 2/(2 - x) + Log[2 - x]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \frac{1-4 x^2+x^3}{(-2+x)^4} \, dx &=\int \left (-\frac{7}{(-2+x)^4}-\frac{4}{(-2+x)^3}+\frac{2}{(-2+x)^2}+\frac{1}{-2+x}\right ) \, dx\\ &=-\frac{7}{3 (2-x)^3}+\frac{2}{(2-x)^2}+\frac{2}{2-x}+\log (2-x)\\ \end{align*}

Mathematica [A]  time = 0.0141903, size = 24, normalized size = 0.67 \[ \frac{-6 x^2+30 x-29}{3 (x-2)^3}+\log (x-2) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 4*x^2 + x^3)/(-2 + x)^4,x]

[Out]

(-29 + 30*x - 6*x^2)/(3*(-2 + x)^3) + Log[-2 + x]

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Maple [A]  time = 0.005, size = 27, normalized size = 0.8 \begin{align*} 2\, \left ( -2+x \right ) ^{-2}+{\frac{7}{3\, \left ( -2+x \right ) ^{3}}}-2\, \left ( -2+x \right ) ^{-1}+\ln \left ( -2+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-4*x^2+1)/(-2+x)^4,x)

[Out]

2/(-2+x)^2+7/3/(-2+x)^3-2/(-2+x)+ln(-2+x)

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Maxima [A]  time = 0.927856, size = 43, normalized size = 1.19 \begin{align*} -\frac{6 \, x^{2} - 30 \, x + 29}{3 \,{\left (x^{3} - 6 \, x^{2} + 12 \, x - 8\right )}} + \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4*x^2+1)/(-2+x)^4,x, algorithm="maxima")

[Out]

-1/3*(6*x^2 - 30*x + 29)/(x^3 - 6*x^2 + 12*x - 8) + log(x - 2)

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Fricas [A]  time = 1.99107, size = 123, normalized size = 3.42 \begin{align*} -\frac{6 \, x^{2} - 3 \,{\left (x^{3} - 6 \, x^{2} + 12 \, x - 8\right )} \log \left (x - 2\right ) - 30 \, x + 29}{3 \,{\left (x^{3} - 6 \, x^{2} + 12 \, x - 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4*x^2+1)/(-2+x)^4,x, algorithm="fricas")

[Out]

-1/3*(6*x^2 - 3*(x^3 - 6*x^2 + 12*x - 8)*log(x - 2) - 30*x + 29)/(x^3 - 6*x^2 + 12*x - 8)

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Sympy [A]  time = 0.108621, size = 29, normalized size = 0.81 \begin{align*} - \frac{6 x^{2} - 30 x + 29}{3 x^{3} - 18 x^{2} + 36 x - 24} + \log{\left (x - 2 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-4*x**2+1)/(-2+x)**4,x)

[Out]

-(6*x**2 - 30*x + 29)/(3*x**3 - 18*x**2 + 36*x - 24) + log(x - 2)

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Giac [A]  time = 1.05217, size = 31, normalized size = 0.86 \begin{align*} -\frac{6 \, x^{2} - 30 \, x + 29}{3 \,{\left (x - 2\right )}^{3}} + \log \left ({\left | x - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4*x^2+1)/(-2+x)^4,x, algorithm="giac")

[Out]

-1/3*(6*x^2 - 30*x + 29)/(x - 2)^3 + log(abs(x - 2))