3.2398 \(\int \frac{(5-x) (3+2 x)^2}{\left (2+5 x+3 x^2\right )^3} \, dx\)

Optimal. Leaf size=57 \[ -\frac{587 x+533}{18 \left (3 x^2+5 x+2\right )^2}+\frac{9918 x+8269}{18 \left (3 x^2+5 x+2\right )}-551 \log (x+1)+551 \log (3 x+2) \]

[Out]

-(533 + 587*x)/(18*(2 + 5*x + 3*x^2)^2) + (8269 + 9918*x)/(18*(2 + 5*x + 3*x^2))
- 551*Log[1 + x] + 551*Log[2 + 3*x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0948321, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{587 x+533}{18 \left (3 x^2+5 x+2\right )^2}+\frac{9918 x+8269}{18 \left (3 x^2+5 x+2\right )}-551 \log (x+1)+551 \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

-(533 + 587*x)/(18*(2 + 5*x + 3*x^2)^2) + (8269 + 9918*x)/(18*(2 + 5*x + 3*x^2))
- 551*Log[1 + x] + 551*Log[2 + 3*x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.3402, size = 53, normalized size = 0.93 \[ - \frac{\left (2 x + 3\right ) \left (139 x + 121\right )}{6 \left (3 x^{2} + 5 x + 2\right )^{2}} + \frac{3306 x + 2849}{6 \left (3 x^{2} + 5 x + 2\right )} - 551 \log{\left (x + 1 \right )} + 551 \log{\left (3 x + 2 \right )} \]

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)**3,x)

[Out]

-(2*x + 3)*(139*x + 121)/(6*(3*x**2 + 5*x + 2)**2) + (3306*x + 2849)/(6*(3*x**2
+ 5*x + 2)) - 551*log(x + 1) + 551*log(3*x + 2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0522571, size = 56, normalized size = 0.98 \[ -\frac{587 x+533}{18 \left (3 x^2+5 x+2\right )^2}+\frac{9918 x+8269}{54 x^2+90 x+36}+551 \log (-6 x-4)-551 \log (-2 (x+1)) \]

Antiderivative was successfully verified.

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

[Out]

-(533 + 587*x)/(18*(2 + 5*x + 3*x^2)^2) + (8269 + 9918*x)/(36 + 90*x + 54*x^2) +
551*Log[-4 - 6*x] - 551*Log[-2*(1 + x)]

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 48, normalized size = 0.8 \[ -{\frac{425}{6\, \left ( 2+3\,x \right ) ^{2}}}+320\, \left ( 2+3\,x \right ) ^{-1}+551\,\ln \left ( 2+3\,x \right ) +3\, \left ( 1+x \right ) ^{-2}+77\, \left ( 1+x \right ) ^{-1}-551\,\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-425/6/(2+3*x)^2+320/(2+3*x)+551*ln(2+3*x)+3/(1+x)^2+77/(1+x)-551*ln(1+x)

_______________________________________________________________________________________

Maxima [A]  time = 0.69308, size = 73, normalized size = 1.28 \[ \frac{9918 \, x^{3} + 24799 \, x^{2} + 20198 \, x + 5335}{6 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + 551 \, \log \left (3 \, x + 2\right ) - 551 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(9918*x^3 + 24799*x^2 + 20198*x + 5335)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)
+ 551*log(3*x + 2) - 551*log(x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.260299, size = 126, normalized size = 2.21 \[ \frac{9918 \, x^{3} + 24799 \, x^{2} + 3306 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 3306 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) + 20198 \, x + 5335}{6 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(9918*x^3 + 24799*x^2 + 3306*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(3*x +
2) - 3306*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(x + 1) + 20198*x + 5335)/(9*x
^4 + 30*x^3 + 37*x^2 + 20*x + 4)

_______________________________________________________________________________________

Sympy [A]  time = 0.472991, size = 49, normalized size = 0.86 \[ \frac{9918 x^{3} + 24799 x^{2} + 20198 x + 5335}{54 x^{4} + 180 x^{3} + 222 x^{2} + 120 x + 24} + 551 \log{\left (x + \frac{2}{3} \right )} - 551 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(9918*x**3 + 24799*x**2 + 20198*x + 5335)/(54*x**4 + 180*x**3 + 222*x**2 + 120*x
+ 24) + 551*log(x + 2/3) - 551*log(x + 1)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.268374, size = 62, normalized size = 1.09 \[ \frac{9918 \, x^{3} + 24799 \, x^{2} + 20198 \, x + 5335}{6 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{2}} + 551 \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - 551 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(9918*x^3 + 24799*x^2 + 20198*x + 5335)/(3*x^2 + 5*x + 2)^2 + 551*ln(abs(3*x
+ 2)) - 551*ln(abs(x + 1))