3.2399 \(\int \frac{(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^3} \, dx\)

Optimal. Leaf size=57 \[ \frac{421 (6 x+5)}{6 \left (3 x^2+5 x+2\right )}-\frac{139 x+121}{6 \left (3 x^2+5 x+2\right )^2}-421 \log (x+1)+421 \log (3 x+2) \]

[Out]

-(121 + 139*x)/(6*(2 + 5*x + 3*x^2)^2) + (421*(5 + 6*x))/(6*(2 + 5*x + 3*x^2)) -
421*Log[1 + x] + 421*Log[2 + 3*x]

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Rubi [A]  time = 0.0545347, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{421 (6 x+5)}{6 \left (3 x^2+5 x+2\right )}-\frac{139 x+121}{6 \left (3 x^2+5 x+2\right )^2}-421 \log (x+1)+421 \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

-(121 + 139*x)/(6*(2 + 5*x + 3*x^2)^2) + (421*(5 + 6*x))/(6*(2 + 5*x + 3*x^2)) -
421*Log[1 + x] + 421*Log[2 + 3*x]

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Rubi in Sympy [A]  time = 9.28122, size = 48, normalized size = 0.84 \[ \frac{421 \left (6 x + 5\right )}{6 \left (3 x^{2} + 5 x + 2\right )} - \frac{139 x + 121}{6 \left (3 x^{2} + 5 x + 2\right )^{2}} - 421 \log{\left (x + 1 \right )} + 421 \log{\left (3 x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

421*(6*x + 5)/(6*(3*x**2 + 5*x + 2)) - (139*x + 121)/(6*(3*x**2 + 5*x + 2)**2) -
421*log(x + 1) + 421*log(3*x + 2)

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Mathematica [A]  time = 0.0340507, size = 57, normalized size = 1. \[ \frac{421 (6 x+5)}{6 \left (3 x^2+5 x+2\right )}-\frac{139 x+121}{6 \left (3 x^2+5 x+2\right )^2}-421 \log (x+1)+421 \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

-(121 + 139*x)/(6*(2 + 5*x + 3*x^2)^2) + (421*(5 + 6*x))/(6*(2 + 5*x + 3*x^2)) -
421*Log[1 + x] + 421*Log[2 + 3*x]

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Maple [A]  time = 0.013, size = 48, normalized size = 0.8 \[ -{\frac{85}{2\, \left ( 2+3\,x \right ) ^{2}}}+226\, \left ( 2+3\,x \right ) ^{-1}+421\,\ln \left ( 2+3\,x \right ) +3\, \left ( 1+x \right ) ^{-2}+65\, \left ( 1+x \right ) ^{-1}-421\,\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-85/2/(2+3*x)^2+226/(2+3*x)+421*ln(2+3*x)+3/(1+x)^2+65/(1+x)-421*ln(1+x)

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Maxima [A]  time = 0.687876, size = 73, normalized size = 1.28 \[ \frac{2526 \, x^{3} + 6315 \, x^{2} + 5146 \, x + 1363}{2 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + 421 \, \log \left (3 \, x + 2\right ) - 421 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2526*x^3 + 6315*x^2 + 5146*x + 1363)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4) +
421*log(3*x + 2) - 421*log(x + 1)

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Fricas [A]  time = 0.26217, size = 126, normalized size = 2.21 \[ \frac{2526 \, x^{3} + 6315 \, x^{2} + 842 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 842 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) + 5146 \, x + 1363}{2 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2526*x^3 + 6315*x^2 + 842*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(3*x + 2)
- 842*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(x + 1) + 5146*x + 1363)/(9*x^4 +
30*x^3 + 37*x^2 + 20*x + 4)

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Sympy [A]  time = 0.443712, size = 49, normalized size = 0.86 \[ \frac{2526 x^{3} + 6315 x^{2} + 5146 x + 1363}{18 x^{4} + 60 x^{3} + 74 x^{2} + 40 x + 8} + 421 \log{\left (x + \frac{2}{3} \right )} - 421 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2526*x**3 + 6315*x**2 + 5146*x + 1363)/(18*x**4 + 60*x**3 + 74*x**2 + 40*x + 8)
+ 421*log(x + 2/3) - 421*log(x + 1)

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GIAC/XCAS [A]  time = 0.272534, size = 62, normalized size = 1.09 \[ \frac{2526 \, x^{3} + 6315 \, x^{2} + 5146 \, x + 1363}{2 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{2}} + 421 \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - 421 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2526*x^3 + 6315*x^2 + 5146*x + 1363)/(3*x^2 + 5*x + 2)^2 + 421*ln(abs(3*x +
2)) - 421*ln(abs(x + 1))