3.2388 \(\int \frac{(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=38 \[ -\frac{587 x+533}{9 \left (3 x^2+5 x+2\right )}+59 \log (x+1)-\frac{535}{9} \log (3 x+2) \]

[Out]

-(533 + 587*x)/(9*(2 + 5*x + 3*x^2)) + 59*Log[1 + x] - (535*Log[2 + 3*x])/9

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Rubi [A]  time = 0.0744933, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{587 x+533}{9 \left (3 x^2+5 x+2\right )}+59 \log (x+1)-\frac{535}{9} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

-(533 + 587*x)/(9*(2 + 5*x + 3*x^2)) + 59*Log[1 + x] - (535*Log[2 + 3*x])/9

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Rubi in Sympy [A]  time = 13.7289, size = 36, normalized size = 0.95 \[ - \frac{\left (2 x + 3\right ) \left (139 x + 121\right )}{3 \left (3 x^{2} + 5 x + 2\right )} + 59 \log{\left (x + 1 \right )} - \frac{535 \log{\left (3 x + 2 \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)*(139*x + 121)/(3*(3*x**2 + 5*x + 2)) + 59*log(x + 1) - 535*log(3*x +
2)/9

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Mathematica [A]  time = 0.0463812, size = 38, normalized size = 1. \[ -\frac{587 x+533}{27 x^2+45 x+18}-\frac{535}{9} \log (-6 x-4)+59 \log (-2 (x+1)) \]

Antiderivative was successfully verified.

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

[Out]

-((533 + 587*x)/(18 + 45*x + 27*x^2)) - (535*Log[-4 - 6*x])/9 + 59*Log[-2*(1 + x
)]

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Maple [A]  time = 0.014, size = 32, normalized size = 0.8 \[ -{\frac{425}{18+27\,x}}-{\frac{535\,\ln \left ( 2+3\,x \right ) }{9}}-6\, \left ( 1+x \right ) ^{-1}+59\,\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-425/9/(2+3*x)-535/9*ln(2+3*x)-6/(1+x)+59*ln(1+x)

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Maxima [A]  time = 0.690425, size = 46, normalized size = 1.21 \[ -\frac{587 \, x + 533}{9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} - \frac{535}{9} \, \log \left (3 \, x + 2\right ) + 59 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(587*x + 533)/(3*x^2 + 5*x + 2) - 535/9*log(3*x + 2) + 59*log(x + 1)

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Fricas [A]  time = 0.262593, size = 72, normalized size = 1.89 \[ -\frac{535 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 531 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 587 \, x + 533}{9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(535*(3*x^2 + 5*x + 2)*log(3*x + 2) - 531*(3*x^2 + 5*x + 2)*log(x + 1) + 58
7*x + 533)/(3*x^2 + 5*x + 2)

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Sympy [A]  time = 0.397527, size = 31, normalized size = 0.82 \[ - \frac{587 x + 533}{27 x^{2} + 45 x + 18} - \frac{535 \log{\left (x + \frac{2}{3} \right )}}{9} + 59 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(587*x + 533)/(27*x**2 + 45*x + 18) - 535*log(x + 2/3)/9 + 59*log(x + 1)

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GIAC/XCAS [A]  time = 0.387974, size = 49, normalized size = 1.29 \[ -\frac{587 \, x + 533}{9 \,{\left (3 \, x + 2\right )}{\left (x + 1\right )}} - \frac{535}{9} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + 59 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(587*x + 533)/((3*x + 2)*(x + 1)) - 535/9*ln(abs(3*x + 2)) + 59*ln(abs(x +
1))