3.267 \(\int \frac{-2-3 x+5 x^2}{(-2+x) x^2} \, dx\)

Optimal. Leaf size=18 \[ -\frac{1}{x}+3 \log (2-x)+2 \log (x) \]

[Out]

-x^(-1) + 3*Log[2 - x] + 2*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0365555, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ -\frac{1}{x}+3 \log (2-x)+2 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(-2 - 3*x + 5*x^2)/((-2 + x)*x^2),x]

[Out]

-x^(-1) + 3*Log[2 - x] + 2*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.81582, size = 14, normalized size = 0.78 \[ 2 \log{\left (x \right )} + 3 \log{\left (- x + 2 \right )} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**2-3*x-2)/(-2+x)/x**2,x)

[Out]

2*log(x) + 3*log(-x + 2) - 1/x

_______________________________________________________________________________________

Mathematica [A]  time = 0.00601856, size = 18, normalized size = 1. \[ -\frac{1}{x}+3 \log (2-x)+2 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(-2 - 3*x + 5*x^2)/((-2 + x)*x^2),x]

[Out]

-x^(-1) + 3*Log[2 - x] + 2*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 17, normalized size = 0.9 \[ -{x}^{-1}+2\,\ln \left ( x \right ) +3\,\ln \left ( -2+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2-3*x-2)/(-2+x)/x^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.41041, size = 22, normalized size = 1.22 \[ -\frac{1}{x} + 3 \, \log \left (x - 2\right ) + 2 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 - 3*x - 2)/((x - 2)*x^2),x, algorithm="maxima")

[Out]

-1/x + 3*log(x - 2) + 2*log(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.204945, size = 24, normalized size = 1.33 \[ \frac{3 \, x \log \left (x - 2\right ) + 2 \, x \log \left (x\right ) - 1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 - 3*x - 2)/((x - 2)*x^2),x, algorithm="fricas")

[Out]

(3*x*log(x - 2) + 2*x*log(x) - 1)/x

_______________________________________________________________________________________

Sympy [A]  time = 0.113655, size = 14, normalized size = 0.78 \[ 2 \log{\left (x \right )} + 3 \log{\left (x - 2 \right )} - \frac{1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2-3*x-2)/(-2+x)/x**2,x)

[Out]

2*log(x) + 3*log(x - 2) - 1/x

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205338, size = 24, normalized size = 1.33 \[ -\frac{1}{x} + 3 \,{\rm ln}\left ({\left | x - 2 \right |}\right ) + 2 \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 - 3*x - 2)/((x - 2)*x^2),x, algorithm="giac")

[Out]

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