3.2387 \(\int \frac{(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=43 \[ -\frac{2611 x+2449}{27 \left (3 x^2+5 x+2\right )}-\frac{8 x}{9}+71 \log (x+1)-\frac{1825}{27} \log (3 x+2) \]

[Out]

(-8*x)/9 - (2449 + 2611*x)/(27*(2 + 5*x + 3*x^2)) + 71*Log[1 + x] - (1825*Log[2
+ 3*x])/27

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Rubi [A]  time = 0.094482, antiderivative size = 43, 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 \[ -\frac{2611 x+2449}{27 \left (3 x^2+5 x+2\right )}-\frac{8 x}{9}+71 \log (x+1)-\frac{1825}{27} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

(-8*x)/9 - (2449 + 2611*x)/(27*(2 + 5*x + 3*x^2)) + 71*Log[1 + x] - (1825*Log[2
+ 3*x])/27

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\left (2 x + 3\right )^{2} \left (139 x + 121\right )}{3 \left (3 x^{2} + 5 x + 2\right )} + 71 \log{\left (x + 1 \right )} - \frac{1825 \log{\left (3 x + 2 \right )}}{27} + \frac{\int \frac{548}{3}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)**2*(139*x + 121)/(3*(3*x**2 + 5*x + 2)) + 71*log(x + 1) - 1825*log(3*
x + 2)/27 + Integral(548/3, x)/3

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Mathematica [A]  time = 0.0611932, size = 47, normalized size = 1.09 \[ -\frac{2611 x+2449}{81 x^2+135 x+54}-\frac{4}{9} (2 x+3)-\frac{1825}{27} \log (-6 x-4)+71 \log (-2 (x+1)) \]

Antiderivative was successfully verified.

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

[Out]

(-4*(3 + 2*x))/9 - (2449 + 2611*x)/(54 + 135*x + 81*x^2) - (1825*Log[-4 - 6*x])/
27 + 71*Log[-2*(1 + x)]

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Maple [A]  time = 0.013, size = 35, normalized size = 0.8 \[ -{\frac{8\,x}{9}}-{\frac{2125}{54+81\,x}}-{\frac{1825\,\ln \left ( 2+3\,x \right ) }{27}}-6\, \left ( 1+x \right ) ^{-1}+71\,\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/9*x-2125/27/(2+3*x)-1825/27*ln(2+3*x)-6/(1+x)+71*ln(1+x)

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Maxima [A]  time = 0.692168, size = 50, normalized size = 1.16 \[ -\frac{8}{9} \, x - \frac{2611 \, x + 2449}{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} - \frac{1825}{27} \, \log \left (3 \, x + 2\right ) + 71 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/9*x - 1/27*(2611*x + 2449)/(3*x^2 + 5*x + 2) - 1825/27*log(3*x + 2) + 71*log(
x + 1)

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Fricas [A]  time = 0.265181, size = 85, normalized size = 1.98 \[ -\frac{72 \, x^{3} + 120 \, x^{2} + 1825 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 1917 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 2659 \, x + 2449}{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(72*x^3 + 120*x^2 + 1825*(3*x^2 + 5*x + 2)*log(3*x + 2) - 1917*(3*x^2 + 5*
x + 2)*log(x + 1) + 2659*x + 2449)/(3*x^2 + 5*x + 2)

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Sympy [A]  time = 0.395492, size = 36, normalized size = 0.84 \[ - \frac{8 x}{9} - \frac{2611 x + 2449}{81 x^{2} + 135 x + 54} - \frac{1825 \log{\left (x + \frac{2}{3} \right )}}{27} + 71 \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*x/9 - (2611*x + 2449)/(81*x**2 + 135*x + 54) - 1825*log(x + 2/3)/27 + 71*log(
x + 1)

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GIAC/XCAS [A]  time = 0.381487, size = 53, normalized size = 1.23 \[ -\frac{8}{9} \, x - \frac{2611 \, x + 2449}{27 \,{\left (3 \, x + 2\right )}{\left (x + 1\right )}} - \frac{1825}{27} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + 71 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/9*x - 1/27*(2611*x + 2449)/((3*x + 2)*(x + 1)) - 1825/27*ln(abs(3*x + 2)) + 7
1*ln(abs(x + 1))