3.2386 \(\int \frac{(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=50 \[ -\frac{8 x^2}{9}-\frac{12083 x+11597}{81 \left (3 x^2+5 x+2\right )}+\frac{112 x}{27}+83 \log (x+1)-\frac{1625}{27} \log (3 x+2) \]

[Out]

(112*x)/27 - (8*x^2)/9 - (11597 + 12083*x)/(81*(2 + 5*x + 3*x^2)) + 83*Log[1 + x
] - (1625*Log[2 + 3*x])/27

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Rubi [A]  time = 0.104572, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{8 x^2}{9}-\frac{12083 x+11597}{81 \left (3 x^2+5 x+2\right )}+\frac{112 x}{27}+83 \log (x+1)-\frac{1625}{27} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(3 + 2*x)^4)/(2 + 5*x + 3*x^2)^2,x]

[Out]

(112*x)/27 - (8*x^2)/9 - (11597 + 12083*x)/(81*(2 + 5*x + 3*x^2)) + 83*Log[1 + x
] - (1625*Log[2 + 3*x])/27

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\left (2 x + 3\right )^{3} \left (139 x + 121\right )}{3 \left (3 x^{2} + 5 x + 2\right )} + 83 \log{\left (x + 1 \right )} - \frac{1625 \log{\left (3 x + 2 \right )}}{27} + \frac{\int \frac{4156}{3}\, dx}{3} + \frac{736 \int x\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**4/(3*x**2+5*x+2)**2,x)

[Out]

-(2*x + 3)**3*(139*x + 121)/(3*(3*x**2 + 5*x + 2)) + 83*log(x + 1) - 1625*log(3*
x + 2)/27 + Integral(4156/3, x)/3 + 736*Integral(x, x)/3

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Mathematica [A]  time = 0.0626341, size = 56, normalized size = 1.12 \[ \frac{1}{81} \left (-\frac{12083 x+11597}{3 x^2+5 x+2}-18 (2 x+3)^2+276 (2 x+3)-4875 \log (-6 x-4)+6723 \log (-2 (x+1))\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(3 + 2*x)^4)/(2 + 5*x + 3*x^2)^2,x]

[Out]

(276*(3 + 2*x) - 18*(3 + 2*x)^2 - (11597 + 12083*x)/(2 + 5*x + 3*x^2) - 4875*Log
[-4 - 6*x] + 6723*Log[-2*(1 + x)])/81

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Maple [A]  time = 0.014, size = 40, normalized size = 0.8 \[ -{\frac{8\,{x}^{2}}{9}}+{\frac{112\,x}{27}}-{\frac{10625}{162+243\,x}}-{\frac{1625\,\ln \left ( 2+3\,x \right ) }{27}}-6\, \left ( 1+x \right ) ^{-1}+83\,\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3+2*x)^4/(3*x^2+5*x+2)^2,x)

[Out]

-8/9*x^2+112/27*x-10625/81/(2+3*x)-1625/27*ln(2+3*x)-6/(1+x)+83*ln(1+x)

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Maxima [A]  time = 0.700993, size = 57, normalized size = 1.14 \[ -\frac{8}{9} \, x^{2} + \frac{112}{27} \, x - \frac{12083 \, x + 11597}{81 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} - \frac{1625}{27} \, \log \left (3 \, x + 2\right ) + 83 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/(3*x^2 + 5*x + 2)^2,x, algorithm="maxima")

[Out]

-8/9*x^2 + 112/27*x - 1/81*(12083*x + 11597)/(3*x^2 + 5*x + 2) - 1625/27*log(3*x
+ 2) + 83*log(x + 1)

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Fricas [A]  time = 0.261239, size = 92, normalized size = 1.84 \[ -\frac{216 \, x^{4} - 648 \, x^{3} - 1536 \, x^{2} + 4875 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 6723 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 11411 \, x + 11597}{81 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/(3*x^2 + 5*x + 2)^2,x, algorithm="fricas")

[Out]

-1/81*(216*x^4 - 648*x^3 - 1536*x^2 + 4875*(3*x^2 + 5*x + 2)*log(3*x + 2) - 6723
*(3*x^2 + 5*x + 2)*log(x + 1) + 11411*x + 11597)/(3*x^2 + 5*x + 2)

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Sympy [A]  time = 0.410545, size = 42, normalized size = 0.84 \[ - \frac{8 x^{2}}{9} + \frac{112 x}{27} - \frac{12083 x + 11597}{243 x^{2} + 405 x + 162} - \frac{1625 \log{\left (x + \frac{2}{3} \right )}}{27} + 83 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3+2*x)**4/(3*x**2+5*x+2)**2,x)

[Out]

-8*x**2/9 + 112*x/27 - (12083*x + 11597)/(243*x**2 + 405*x + 162) - 1625*log(x +
2/3)/27 + 83*log(x + 1)

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GIAC/XCAS [A]  time = 0.401067, size = 59, normalized size = 1.18 \[ -\frac{8}{9} \, x^{2} + \frac{112}{27} \, x - \frac{12083 \, x + 11597}{81 \,{\left (3 \, x + 2\right )}{\left (x + 1\right )}} - \frac{1625}{27} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + 83 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^4*(x - 5)/(3*x^2 + 5*x + 2)^2,x, algorithm="giac")

[Out]

-8/9*x^2 + 112/27*x - 1/81*(12083*x + 11597)/((3*x + 2)*(x + 1)) - 1625/27*ln(ab
s(3*x + 2)) + 83*ln(abs(x + 1))