3.2249 \(\int \frac{x^2}{2-3 x+x^2} \, dx\)

Optimal. Leaf size=18 \[ x-\log (1-x)+4 \log (2-x) \]

[Out]

x - Log[1 - x] + 4*Log[2 - x]

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Rubi [A]  time = 0.0308976, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ x-\log (1-x)+4 \log (2-x) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(2 - 3*x + x^2),x]

[Out]

x - Log[1 - x] + 4*Log[2 - x]

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Rubi in Sympy [A]  time = 6.55147, size = 12, normalized size = 0.67 \[ x - \log{\left (- x + 1 \right )} + 4 \log{\left (- x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - log(-x + 1) + 4*log(-x + 2)

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Mathematica [A]  time = 0.00679068, size = 18, normalized size = 1. \[ x-\log (1-x)+4 \log (2-x) \]

Antiderivative was successfully verified.

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

[Out]

x - Log[1 - x] + 4*Log[2 - x]

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Maple [A]  time = 0.008, size = 15, normalized size = 0.8 \[ x-\ln \left ( -1+x \right ) +4\,\ln \left ( x-2 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^2-3*x+2),x)

[Out]

x-ln(-1+x)+4*ln(x-2)

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Maxima [A]  time = 0.669705, size = 19, normalized size = 1.06 \[ x - \log \left (x - 1\right ) + 4 \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - log(x - 1) + 4*log(x - 2)

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Fricas [A]  time = 0.201707, size = 19, normalized size = 1.06 \[ x - \log \left (x - 1\right ) + 4 \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - log(x - 1) + 4*log(x - 2)

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Sympy [A]  time = 0.197611, size = 12, normalized size = 0.67 \[ x + 4 \log{\left (x - 2 \right )} - \log{\left (x - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**2-3*x+2),x)

[Out]

x + 4*log(x - 2) - log(x - 1)

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GIAC/XCAS [A]  time = 0.203858, size = 22, normalized size = 1.22 \[ x -{\rm ln}\left ({\left | x - 1 \right |}\right ) + 4 \,{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x - ln(abs(x - 1)) + 4*ln(abs(x - 2))