3.2382 \(\int \frac{5-x}{(3+2 x) \left (2+5 x+3 x^2\right )} \, dx\)

Optimal. Leaf size=27 \[ -6 \log (x+1)+\frac{13}{5} \log (2 x+3)+\frac{17}{5} \log (3 x+2) \]

[Out]

-6*Log[1 + x] + (13*Log[3 + 2*x])/5 + (17*Log[2 + 3*x])/5

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Rubi [A]  time = 0.0567234, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ -6 \log (x+1)+\frac{13}{5} \log (2 x+3)+\frac{17}{5} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

-6*Log[1 + x] + (13*Log[3 + 2*x])/5 + (17*Log[2 + 3*x])/5

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Rubi in Sympy [A]  time = 12.0224, size = 26, normalized size = 0.96 \[ - 6 \log{\left (x + 1 \right )} + \frac{13 \log{\left (2 x + 3 \right )}}{5} + \frac{17 \log{\left (3 x + 2 \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*log(x + 1) + 13*log(2*x + 3)/5 + 17*log(3*x + 2)/5

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Mathematica [A]  time = 0.0137186, size = 27, normalized size = 1. \[ -6 \log (x+1)+\frac{13}{5} \log (2 x+3)+\frac{17}{5} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

-6*Log[1 + x] + (13*Log[3 + 2*x])/5 + (17*Log[2 + 3*x])/5

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Maple [A]  time = 0.01, size = 24, normalized size = 0.9 \[ -6\,\ln \left ( 1+x \right ) +{\frac{13\,\ln \left ( 3+2\,x \right ) }{5}}+{\frac{17\,\ln \left ( 2+3\,x \right ) }{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*ln(1+x)+13/5*ln(3+2*x)+17/5*ln(2+3*x)

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Maxima [A]  time = 0.688214, size = 31, normalized size = 1.15 \[ \frac{17}{5} \, \log \left (3 \, x + 2\right ) + \frac{13}{5} \, \log \left (2 \, x + 3\right ) - 6 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17/5*log(3*x + 2) + 13/5*log(2*x + 3) - 6*log(x + 1)

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Fricas [A]  time = 0.268311, size = 31, normalized size = 1.15 \[ \frac{17}{5} \, \log \left (3 \, x + 2\right ) + \frac{13}{5} \, \log \left (2 \, x + 3\right ) - 6 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17/5*log(3*x + 2) + 13/5*log(2*x + 3) - 6*log(x + 1)

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Sympy [A]  time = 0.310149, size = 26, normalized size = 0.96 \[ \frac{17 \log{\left (x + \frac{2}{3} \right )}}{5} - 6 \log{\left (x + 1 \right )} + \frac{13 \log{\left (x + \frac{3}{2} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17*log(x + 2/3)/5 - 6*log(x + 1) + 13*log(x + 3/2)/5

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GIAC/XCAS [A]  time = 0.301961, size = 35, normalized size = 1.3 \[ \frac{17}{5} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{13}{5} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - 6 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17/5*ln(abs(3*x + 2)) + 13/5*ln(abs(2*x + 3)) - 6*ln(abs(x + 1))