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

Optimal. Leaf size=22 \[ \frac{5}{1-x}-\log (1-x)+3 \log (x) \]

[Out]

5/(1 - x) - Log[1 - x] + 3*Log[x]

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Rubi [A]  time = 0.0420829, antiderivative size = 22, 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 \[ \frac{5}{1-x}-\log (1-x)+3 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

5/(1 - x) - Log[1 - x] + 3*Log[x]

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Rubi in Sympy [A]  time = 4.40424, size = 14, normalized size = 0.64 \[ 3 \log{\left (x \right )} - \log{\left (- x + 1 \right )} + \frac{5}{- x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(x) - log(-x + 1) + 5/(-x + 1)

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Mathematica [A]  time = 0.0132863, size = 20, normalized size = 0.91 \[ -\frac{5}{x-1}-\log (1-x)+3 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

-5/(-1 + x) - Log[1 - x] + 3*Log[x]

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Maple [A]  time = 0.011, size = 19, normalized size = 0.9 \[ -5\, \left ( -1+x \right ) ^{-1}-\ln \left ( -1+x \right ) +3\,\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^2+3)/(-1+x)^2/x,x)

[Out]

-5/(-1+x)-ln(-1+x)+3*ln(x)

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Maxima [A]  time = 0.944092, size = 24, normalized size = 1.09 \[ -\frac{5}{x - 1} - \log \left (x - 1\right ) + 3 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/(x - 1) - log(x - 1) + 3*log(x)

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Fricas [A]  time = 0.244985, size = 32, normalized size = 1.45 \[ -\frac{{\left (x - 1\right )} \log \left (x - 1\right ) - 3 \,{\left (x - 1\right )} \log \left (x\right ) + 5}{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-((x - 1)*log(x - 1) - 3*(x - 1)*log(x) + 5)/(x - 1)

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Sympy [A]  time = 0.225205, size = 14, normalized size = 0.64 \[ 3 \log{\left (x \right )} - \log{\left (x - 1 \right )} - \frac{5}{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(x) - log(x - 1) - 5/(x - 1)

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GIAC/XCAS [A]  time = 0.259594, size = 38, normalized size = 1.73 \[ -\frac{5}{x - 1} + 2 \,{\rm ln}\left ({\left | x - 1 \right |}\right ) + 3 \,{\rm ln}\left ({\left | -\frac{1}{x - 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/(x - 1) + 2*ln(abs(x - 1)) + 3*ln(abs(-1/(x - 1) - 1))