3.2254 \(\int \frac{x^3}{1-2 x+x^2} \, dx\)

Optimal. Leaf size=26 \[ \frac{x^2}{2}+2 x+\frac{1}{1-x}+3 \log (1-x) \]

[Out]

(1 - x)^(-1) + 2*x + x^2/2 + 3*Log[1 - x]

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Rubi [A]  time = 0.0291117, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{x^2}{2}+2 x+\frac{1}{1-x}+3 \log (1-x) \]

Antiderivative was successfully verified.

[In]  Int[x^3/(1 - 2*x + x^2),x]

[Out]

(1 - x)^(-1) + 2*x + x^2/2 + 3*Log[1 - x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 3 \log{\left (- x + 1 \right )} + 4 \int \frac{1}{2}\, dx + \int x\, dx + \frac{1}{- x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(x**2-2*x+1),x)

[Out]

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

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Mathematica [A]  time = 0.0143132, size = 25, normalized size = 0.96 \[ \frac{1}{2} \left (x^2+4 x-\frac{2}{x-1}+6 \log (x-1)-5\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-5 - 2/(-1 + x) + 4*x + x^2 + 6*Log[-1 + x])/2

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Maple [A]  time = 0.008, size = 23, normalized size = 0.9 \[ 2\,x+{\frac{{x}^{2}}{2}}+3\,\ln \left ( -1+x \right ) - \left ( -1+x \right ) ^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^2-2*x+1),x)

[Out]

2*x+1/2*x^2+3*ln(-1+x)-1/(-1+x)

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Maxima [A]  time = 0.673086, size = 30, normalized size = 1.15 \[ \frac{1}{2} \, x^{2} + 2 \, x - \frac{1}{x - 1} + 3 \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 + 2*x - 1/(x - 1) + 3*log(x - 1)

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Fricas [A]  time = 0.201684, size = 39, normalized size = 1.5 \[ \frac{x^{3} + 3 \, x^{2} + 6 \,{\left (x - 1\right )} \log \left (x - 1\right ) - 4 \, x - 2}{2 \,{\left (x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(x^3 + 3*x^2 + 6*(x - 1)*log(x - 1) - 4*x - 2)/(x - 1)

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Sympy [A]  time = 0.137438, size = 19, normalized size = 0.73 \[ \frac{x^{2}}{2} + 2 x + 3 \log{\left (x - 1 \right )} - \frac{1}{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(x**2-2*x+1),x)

[Out]

x**2/2 + 2*x + 3*log(x - 1) - 1/(x - 1)

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GIAC/XCAS [A]  time = 0.204206, size = 31, normalized size = 1.19 \[ \frac{1}{2} \, x^{2} + 2 \, x - \frac{1}{x - 1} + 3 \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 + 2*x - 1/(x - 1) + 3*ln(abs(x - 1))