3.164 \(\int \frac{-4+3 x+x^2}{(-1+2 x)^2 (3+2 x)} \, dx\)

Optimal. Leaf size=32 \[ -\frac{9}{32 (1-2 x)}+\frac{41}{128} \log (1-2 x)-\frac{25}{128} \log (2 x+3) \]

[Out]

-9/(32*(1 - 2*x)) + (41*Log[1 - 2*x])/128 - (25*Log[3 + 2*x])/128

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Rubi [A]  time = 0.0588609, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{9}{32 (1-2 x)}+\frac{41}{128} \log (1-2 x)-\frac{25}{128} \log (2 x+3) \]

Antiderivative was successfully verified.

[In]  Int[(-4 + 3*x + x^2)/((-1 + 2*x)^2*(3 + 2*x)),x]

[Out]

-9/(32*(1 - 2*x)) + (41*Log[1 - 2*x])/128 - (25*Log[3 + 2*x])/128

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Rubi in Sympy [A]  time = 3.79581, size = 26, normalized size = 0.81 \[ \frac{41 \log{\left (- 2 x + 1 \right )}}{128} - \frac{25 \log{\left (2 x + 3 \right )}}{128} - \frac{9}{32 \left (- 2 x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+3*x-4)/(2*x-1)**2/(3+2*x),x)

[Out]

41*log(-2*x + 1)/128 - 25*log(2*x + 3)/128 - 9/(32*(-2*x + 1))

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Mathematica [A]  time = 0.0241322, size = 32, normalized size = 1. \[ \frac{9}{32 (2 x-1)}+\frac{41}{128} \log (1-2 x)-\frac{25}{128} \log (2 x+3) \]

Antiderivative was successfully verified.

[In]  Integrate[(-4 + 3*x + x^2)/((-1 + 2*x)^2*(3 + 2*x)),x]

[Out]

9/(32*(-1 + 2*x)) + (41*Log[1 - 2*x])/128 - (25*Log[3 + 2*x])/128

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Maple [A]  time = 0.013, size = 27, normalized size = 0.8 \[ -{\frac{25\,\ln \left ( 3+2\,x \right ) }{128}}+{\frac{9}{64\,x-32}}+{\frac{41\,\ln \left ( 2\,x-1 \right ) }{128}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+3*x-4)/(2*x-1)^2/(3+2*x),x)

[Out]

-25/128*ln(3+2*x)+9/32/(2*x-1)+41/128*ln(2*x-1)

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Maxima [A]  time = 1.37685, size = 35, normalized size = 1.09 \[ \frac{9}{32 \,{\left (2 \, x - 1\right )}} - \frac{25}{128} \, \log \left (2 \, x + 3\right ) + \frac{41}{128} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 3*x - 4)/((2*x + 3)*(2*x - 1)^2),x, algorithm="maxima")

[Out]

9/32/(2*x - 1) - 25/128*log(2*x + 3) + 41/128*log(2*x - 1)

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Fricas [A]  time = 0.211184, size = 50, normalized size = 1.56 \[ -\frac{25 \,{\left (2 \, x - 1\right )} \log \left (2 \, x + 3\right ) - 41 \,{\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 36}{128 \,{\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 3*x - 4)/((2*x + 3)*(2*x - 1)^2),x, algorithm="fricas")

[Out]

-1/128*(25*(2*x - 1)*log(2*x + 3) - 41*(2*x - 1)*log(2*x - 1) - 36)/(2*x - 1)

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Sympy [A]  time = 0.146379, size = 26, normalized size = 0.81 \[ \frac{41 \log{\left (x - \frac{1}{2} \right )}}{128} - \frac{25 \log{\left (x + \frac{3}{2} \right )}}{128} + \frac{9}{64 x - 32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+3*x-4)/(2*x-1)**2/(3+2*x),x)

[Out]

41*log(x - 1/2)/128 - 25*log(x + 3/2)/128 + 9/(64*x - 32)

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GIAC/XCAS [A]  time = 0.217145, size = 58, normalized size = 1.81 \[ \frac{9}{32 \,{\left (2 \, x - 1\right )}} - \frac{1}{8} \,{\rm ln}\left (\frac{{\left | 2 \, x - 1 \right |}}{2 \,{\left (2 \, x - 1\right )}^{2}}\right ) - \frac{25}{128} \,{\rm ln}\left ({\left | -\frac{4}{2 \, x - 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 3*x - 4)/((2*x + 3)*(2*x - 1)^2),x, algorithm="giac")

[Out]

9/32/(2*x - 1) - 1/8*ln(1/2*abs(2*x - 1)/(2*x - 1)^2) - 25/128*ln(abs(-4/(2*x -
1) - 1))