3.2400 \(\int \frac{5-x}{\left (2+5 x+3 x^2\right )^3} \, dx\)

Optimal. Leaf size=57 \[ \frac{105 (6 x+5)}{2 \left (3 x^2+5 x+2\right )}-\frac{35 x+29}{2 \left (3 x^2+5 x+2\right )^2}-315 \log (x+1)+315 \log (3 x+2) \]

[Out]

-(29 + 35*x)/(2*(2 + 5*x + 3*x^2)^2) + (105*(5 + 6*x))/(2*(2 + 5*x + 3*x^2)) - 3
15*Log[1 + x] + 315*Log[2 + 3*x]

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Rubi [A]  time = 0.0362947, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{105 (6 x+5)}{2 \left (3 x^2+5 x+2\right )}-\frac{35 x+29}{2 \left (3 x^2+5 x+2\right )^2}-315 \log (x+1)+315 \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

-(29 + 35*x)/(2*(2 + 5*x + 3*x^2)^2) + (105*(5 + 6*x))/(2*(2 + 5*x + 3*x^2)) - 3
15*Log[1 + x] + 315*Log[2 + 3*x]

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Rubi in Sympy [A]  time = 6.2737, size = 48, normalized size = 0.84 \[ \frac{105 \left (6 x + 5\right )}{2 \left (3 x^{2} + 5 x + 2\right )} - \frac{35 x + 29}{2 \left (3 x^{2} + 5 x + 2\right )^{2}} - 315 \log{\left (x + 1 \right )} + 315 \log{\left (3 x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

105*(6*x + 5)/(2*(3*x**2 + 5*x + 2)) - (35*x + 29)/(2*(3*x**2 + 5*x + 2)**2) - 3
15*log(x + 1) + 315*log(3*x + 2)

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Mathematica [A]  time = 0.0284628, size = 57, normalized size = 1. \[ \frac{-35 x-29}{2 \left (3 x^2+5 x+2\right )^2}+\frac{105 (6 x+5)}{2 \left (3 x^2+5 x+2\right )}-315 \log (x+1)+315 \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

(-29 - 35*x)/(2*(2 + 5*x + 3*x^2)^2) + (105*(5 + 6*x))/(2*(2 + 5*x + 3*x^2)) - 3
15*Log[1 + x] + 315*Log[2 + 3*x]

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Maple [A]  time = 0.013, size = 48, normalized size = 0.8 \[ -{\frac{51}{2\, \left ( 2+3\,x \right ) ^{2}}}+156\, \left ( 2+3\,x \right ) ^{-1}+315\,\ln \left ( 2+3\,x \right ) +3\, \left ( 1+x \right ) ^{-2}+53\, \left ( 1+x \right ) ^{-1}-315\,\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)/(3*x^2+5*x+2)^3,x)

[Out]

-51/2/(2+3*x)^2+156/(2+3*x)+315*ln(2+3*x)+3/(1+x)^2+53/(1+x)-315*ln(1+x)

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Maxima [A]  time = 0.686712, size = 73, normalized size = 1.28 \[ \frac{1890 \, x^{3} + 4725 \, x^{2} + 3850 \, x + 1021}{2 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + 315 \, \log \left (3 \, x + 2\right ) - 315 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(1890*x^3 + 4725*x^2 + 3850*x + 1021)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4) +
315*log(3*x + 2) - 315*log(x + 1)

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Fricas [A]  time = 0.262914, size = 126, normalized size = 2.21 \[ \frac{1890 \, x^{3} + 4725 \, x^{2} + 630 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 630 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) + 3850 \, x + 1021}{2 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(1890*x^3 + 4725*x^2 + 630*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(3*x + 2)
- 630*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(x + 1) + 3850*x + 1021)/(9*x^4 +
30*x^3 + 37*x^2 + 20*x + 4)

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Sympy [A]  time = 0.411123, size = 49, normalized size = 0.86 \[ \frac{1890 x^{3} + 4725 x^{2} + 3850 x + 1021}{18 x^{4} + 60 x^{3} + 74 x^{2} + 40 x + 8} + 315 \log{\left (x + \frac{2}{3} \right )} - 315 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(1890*x**3 + 4725*x**2 + 3850*x + 1021)/(18*x**4 + 60*x**3 + 74*x**2 + 40*x + 8)
+ 315*log(x + 2/3) - 315*log(x + 1)

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GIAC/XCAS [A]  time = 0.269714, size = 62, normalized size = 1.09 \[ \frac{1890 \, x^{3} + 4725 \, x^{2} + 3850 \, x + 1021}{2 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{2}} + 315 \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - 315 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(1890*x^3 + 4725*x^2 + 3850*x + 1021)/(3*x^2 + 5*x + 2)^2 + 315*ln(abs(3*x +
2)) - 315*ln(abs(x + 1))