3.2252 \(\int \frac{x^3}{2-3 x+x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{x^2}{2}+3 x-\log (1-x)+8 \log (2-x) \]

[Out]

3*x + x^2/2 - Log[1 - x] + 8*Log[2 - x]

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Rubi [A]  time = 0.0392936, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{x^2}{2}+3 x-\log (1-x)+8 \log (2-x) \]

Antiderivative was successfully verified.

[In]  Int[x^3/(2 - 3*x + x^2),x]

[Out]

3*x + x^2/2 - Log[1 - x] + 8*Log[2 - x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 3 x - \log{\left (- x + 1 \right )} + 8 \log{\left (- x + 2 \right )} + \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x - log(-x + 1) + 8*log(-x + 2) + Integral(x, x)

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Mathematica [A]  time = 0.00688475, size = 27, normalized size = 1. \[ \frac{x^2}{2}+3 x-\log (1-x)+8 \log (2-x) \]

Antiderivative was successfully verified.

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

[Out]

3*x + x^2/2 - Log[1 - x] + 8*Log[2 - x]

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Maple [A]  time = 0.007, size = 22, normalized size = 0.8 \[ 3\,x+{\frac{{x}^{2}}{2}}-\ln \left ( -1+x \right ) +8\,\ln \left ( x-2 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^2-3*x+2),x)

[Out]

3*x+1/2*x^2-ln(-1+x)+8*ln(x-2)

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Maxima [A]  time = 0.677108, size = 28, normalized size = 1.04 \[ \frac{1}{2} \, x^{2} + 3 \, x - \log \left (x - 1\right ) + 8 \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 + 3*x - log(x - 1) + 8*log(x - 2)

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Fricas [A]  time = 0.201441, size = 28, normalized size = 1.04 \[ \frac{1}{2} \, x^{2} + 3 \, x - \log \left (x - 1\right ) + 8 \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 + 3*x - log(x - 1) + 8*log(x - 2)

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Sympy [A]  time = 0.196264, size = 19, normalized size = 0.7 \[ \frac{x^{2}}{2} + 3 x + 8 \log{\left (x - 2 \right )} - \log{\left (x - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(x**2-3*x+2),x)

[Out]

x**2/2 + 3*x + 8*log(x - 2) - log(x - 1)

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GIAC/XCAS [A]  time = 0.204488, size = 31, normalized size = 1.15 \[ \frac{1}{2} \, x^{2} + 3 \, x -{\rm ln}\left ({\left | x - 1 \right |}\right ) + 8 \,{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2 + 3*x - ln(abs(x - 1)) + 8*ln(abs(x - 2))