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

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

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Rubi [A]  time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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IntegrateAlgebraic [A]  time = 0.02, 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]

IntegrateAlgebraic[(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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fricas [A]  time = 1.09, size = 46, normalized size = 1.28 \[ -\frac {6 \, x^{2} - 3 \, {\left (x^{3} - 6 \, x^{2} + 12 \, x - 8\right )} \log \left (x - 2\right ) - 30 \, x + 29}{3 \, {\left (x^{3} - 6 \, x^{2} + 12 \, x - 8\right )}} \]

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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giac [A]  time = 0.85, size = 23, normalized size = 0.64 \[ -\frac {6 \, x^{2} - 30 \, x + 29}{3 \, {\left (x - 2\right )}^{3}} + \log \left ({\left | x - 2 \right |}\right ) \]

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))

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maple [A]  time = 0.29, size = 22, normalized size = 0.61

method	result	size
norman	\(\frac {-2 x^{2}+10 x -\frac {29}{3}}{\left (-2+x \right )^{3}}+\ln \left (-2+x \right )\)	\(22\)
risch	\(\frac {-2 x^{2}+10 x -\frac {29}{3}}{\left (-2+x \right )^{3}}+\ln \left (-2+x \right )\)	\(22\)
default	\(\frac {2}{\left (-2+x \right )^{2}}+\frac {7}{3 \left (-2+x \right )^{3}}+\ln \left (-2+x \right )-\frac {2}{-2+x}\)	\(27\)
meijerg	\(\frac {x \left (\frac {1}{4} x^{2}-\frac {3}{2} x +3\right )}{48 \left (1-\frac {x}{2}\right )^{3}}+\frac {x \left (\frac {11}{2} x^{2}-15 x +12\right )}{24 \left (1-\frac {x}{2}\right )^{3}}+\ln \left (1-\frac {x}{2}\right )-\frac {x^{3}}{12 \left (1-\frac {x}{2}\right )^{3}}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*x^2+10*x-29/3)/(-2+x)^3+ln(-2+x)

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maxima [A]  time = 0.44, size = 32, normalized size = 0.89 \[ -\frac {6 \, x^{2} - 30 \, x + 29}{3 \, {\left (x^{3} - 6 \, x^{2} + 12 \, x - 8\right )}} + \log \left (x - 2\right ) \]

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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mupad [B]  time = 0.04, size = 22, normalized size = 0.61 \[ \ln \left (x-2\right )-\frac {2\,x^2-10\,x+\frac {29}{3}}{{\left (x-2\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 - 4*x^2 + 1)/(x - 2)^4,x)

[Out]

log(x - 2) - (2*x^2 - 10*x + 29/3)/(x - 2)^3

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sympy [A]  time = 0.12, size = 29, normalized size = 0.81 \[ \frac {- 6 x^{2} + 30 x - 29}{3 x^{3} - 18 x^{2} + 36 x - 24} + \log {\left (x - 2 \right )} \]

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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