3.2391 \(\int \frac{5-x}{(3+2 x) \left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=48 \[ -\frac{3 (47 x+37)}{5 \left (3 x^2+5 x+2\right )}+23 \log (x+1)+\frac{52}{25} \log (2 x+3)-\frac{627}{25} \log (3 x+2) \]

[Out]

(-3*(37 + 47*x))/(5*(2 + 5*x + 3*x^2)) + 23*Log[1 + x] + (52*Log[3 + 2*x])/25 -
(627*Log[2 + 3*x])/25

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Rubi [A]  time = 0.0968883, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{3 (47 x+37)}{5 \left (3 x^2+5 x+2\right )}+23 \log (x+1)+\frac{52}{25} \log (2 x+3)-\frac{627}{25} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

(-3*(37 + 47*x))/(5*(2 + 5*x + 3*x^2)) + 23*Log[1 + x] + (52*Log[3 + 2*x])/25 -
(627*Log[2 + 3*x])/25

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Rubi in Sympy [A]  time = 19.7991, size = 41, normalized size = 0.85 \[ - \frac{141 x + 111}{5 \left (3 x^{2} + 5 x + 2\right )} + 23 \log{\left (x + 1 \right )} + \frac{52 \log{\left (2 x + 3 \right )}}{25} - \frac{627 \log{\left (3 x + 2 \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(141*x + 111)/(5*(3*x**2 + 5*x + 2)) + 23*log(x + 1) + 52*log(2*x + 3)/25 - 627
*log(3*x + 2)/25

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Mathematica [A]  time = 0.0575054, size = 48, normalized size = 1. \[ \frac{1}{25} \left (-\frac{15 (47 x+37)}{3 x^2+5 x+2}-627 \log (-6 x-4)+575 \log (-2 (x+1))+52 \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-15*(37 + 47*x))/(2 + 5*x + 3*x^2) - 627*Log[-4 - 6*x] + 575*Log[-2*(1 + x)] +
52*Log[3 + 2*x])/25

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Maple [A]  time = 0.016, size = 40, normalized size = 0.8 \[ -{\frac{51}{10+15\,x}}-{\frac{627\,\ln \left ( 2+3\,x \right ) }{25}}+{\frac{52\,\ln \left ( 3+2\,x \right ) }{25}}-6\, \left ( 1+x \right ) ^{-1}+23\,\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-51/5/(2+3*x)-627/25*ln(2+3*x)+52/25*ln(3+2*x)-6/(1+x)+23*ln(1+x)

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Maxima [A]  time = 0.68962, size = 57, normalized size = 1.19 \[ -\frac{3 \,{\left (47 \, x + 37\right )}}{5 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} - \frac{627}{25} \, \log \left (3 \, x + 2\right ) + \frac{52}{25} \, \log \left (2 \, x + 3\right ) + 23 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/5*(47*x + 37)/(3*x^2 + 5*x + 2) - 627/25*log(3*x + 2) + 52/25*log(2*x + 3) +
23*log(x + 1)

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Fricas [A]  time = 0.266782, size = 96, normalized size = 2. \[ -\frac{627 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 52 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (2 \, x + 3\right ) - 575 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 705 \, x + 555}{25 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/25*(627*(3*x^2 + 5*x + 2)*log(3*x + 2) - 52*(3*x^2 + 5*x + 2)*log(2*x + 3) -
575*(3*x^2 + 5*x + 2)*log(x + 1) + 705*x + 555)/(3*x^2 + 5*x + 2)

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Sympy [A]  time = 0.484166, size = 41, normalized size = 0.85 \[ - \frac{141 x + 111}{15 x^{2} + 25 x + 10} - \frac{627 \log{\left (x + \frac{2}{3} \right )}}{25} + 23 \log{\left (x + 1 \right )} + \frac{52 \log{\left (x + \frac{3}{2} \right )}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(141*x + 111)/(15*x**2 + 25*x + 10) - 627*log(x + 2/3)/25 + 23*log(x + 1) + 52*
log(x + 3/2)/25

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GIAC/XCAS [A]  time = 0.275496, size = 61, normalized size = 1.27 \[ -\frac{3 \,{\left (47 \, x + 37\right )}}{5 \,{\left (3 \, x + 2\right )}{\left (x + 1\right )}} - \frac{627}{25} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{52}{25} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) + 23 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/5*(47*x + 37)/((3*x + 2)*(x + 1)) - 627/25*ln(abs(3*x + 2)) + 52/25*ln(abs(2*
x + 3)) + 23*ln(abs(x + 1))