3.2389 \(\int \frac{(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=36 \[ -\frac{139 x+121}{3 \left (3 x^2+5 x+2\right )}+47 \log (x+1)-47 \log (3 x+2) \]

[Out]

-(121 + 139*x)/(3*(2 + 5*x + 3*x^2)) + 47*Log[1 + x] - 47*Log[2 + 3*x]

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Rubi [A]  time = 0.0448165, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ -\frac{139 x+121}{3 \left (3 x^2+5 x+2\right )}+47 \log (x+1)-47 \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

-(121 + 139*x)/(3*(2 + 5*x + 3*x^2)) + 47*Log[1 + x] - 47*Log[2 + 3*x]

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Rubi in Sympy [A]  time = 8.56751, size = 32, normalized size = 0.89 \[ - \frac{\left (- x + 5\right ) \left (8 x + 7\right )}{3 x^{2} + 5 x + 2} + 47 \log{\left (x + 1 \right )} - 47 \log{\left (3 x + 2 \right )} \]

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]

-(-x + 5)*(8*x + 7)/(3*x**2 + 5*x + 2) + 47*log(x + 1) - 47*log(3*x + 2)

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Mathematica [A]  time = 0.0254025, size = 34, normalized size = 0.94 \[ -\frac{139 x+121}{9 x^2+15 x+6}+47 \log (x+1)-47 \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

-((121 + 139*x)/(6 + 15*x + 9*x^2)) + 47*Log[1 + x] - 47*Log[2 + 3*x]

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Maple [A]  time = 0.015, size = 32, normalized size = 0.9 \[ -{\frac{85}{6+9\,x}}-47\,\ln \left ( 2+3\,x \right ) -6\, \left ( 1+x \right ) ^{-1}+47\,\ln \left ( 1+x \right ) \]

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]

-85/3/(2+3*x)-47*ln(2+3*x)-6/(1+x)+47*ln(1+x)

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Maxima [A]  time = 0.693901, size = 46, normalized size = 1.28 \[ -\frac{139 \, x + 121}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} - 47 \, \log \left (3 \, x + 2\right ) + 47 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(139*x + 121)/(3*x^2 + 5*x + 2) - 47*log(3*x + 2) + 47*log(x + 1)

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Fricas [A]  time = 0.26325, size = 72, normalized size = 2. \[ -\frac{141 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 141 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 139 \, x + 121}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(141*(3*x^2 + 5*x + 2)*log(3*x + 2) - 141*(3*x^2 + 5*x + 2)*log(x + 1) + 13
9*x + 121)/(3*x^2 + 5*x + 2)

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Sympy [A]  time = 0.333921, size = 29, normalized size = 0.81 \[ - \frac{139 x + 121}{9 x^{2} + 15 x + 6} - 47 \log{\left (x + \frac{2}{3} \right )} + 47 \log{\left (x + 1 \right )} \]

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]

-(139*x + 121)/(9*x**2 + 15*x + 6) - 47*log(x + 2/3) + 47*log(x + 1)

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GIAC/XCAS [A]  time = 0.312157, size = 49, normalized size = 1.36 \[ -\frac{139 \, x + 121}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} - 47 \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + 47 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(139*x + 121)/(3*x^2 + 5*x + 2) - 47*ln(abs(3*x + 2)) + 47*ln(abs(x + 1))