3.2401 \(\int \frac{5-x}{(3+2 x) \left (2+5 x+3 x^2\right )^3} \, dx\)

Optimal. Leaf size=69 \[ -\frac{3 (47 x+37)}{10 \left (3 x^2+5 x+2\right )^2}+\frac{11442 x+9587}{50 \left (3 x^2+5 x+2\right )}-233 \log (x+1)+\frac{208}{125} \log (2 x+3)+\frac{28917}{125} \log (3 x+2) \]

[Out]

(-3*(37 + 47*x))/(10*(2 + 5*x + 3*x^2)^2) + (9587 + 11442*x)/(50*(2 + 5*x + 3*x^
2)) - 233*Log[1 + x] + (208*Log[3 + 2*x])/125 + (28917*Log[2 + 3*x])/125

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Rubi [A]  time = 0.13905, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{3 (47 x+37)}{10 \left (3 x^2+5 x+2\right )^2}+\frac{11442 x+9587}{50 \left (3 x^2+5 x+2\right )}-233 \log (x+1)+\frac{208}{125} \log (2 x+3)+\frac{28917}{125} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

(-3*(37 + 47*x))/(10*(2 + 5*x + 3*x^2)^2) + (9587 + 11442*x)/(50*(2 + 5*x + 3*x^
2)) - 233*Log[1 + x] + (208*Log[3 + 2*x])/125 + (28917*Log[2 + 3*x])/125

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Rubi in Sympy [A]  time = 27.8947, size = 60, normalized size = 0.87 \[ - \frac{141 x + 111}{10 \left (3 x^{2} + 5 x + 2\right )^{2}} + \frac{11442 x + 9587}{50 \left (3 x^{2} + 5 x + 2\right )} - 233 \log{\left (x + 1 \right )} + \frac{208 \log{\left (2 x + 3 \right )}}{125} + \frac{28917 \log{\left (3 x + 2 \right )}}{125} \]

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]

-(141*x + 111)/(10*(3*x**2 + 5*x + 2)**2) + (11442*x + 9587)/(50*(3*x**2 + 5*x +
2)) - 233*log(x + 1) + 208*log(2*x + 3)/125 + 28917*log(3*x + 2)/125

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Mathematica [A]  time = 0.0806476, size = 68, normalized size = 0.99 \[ \frac{1}{125} \left (-\frac{75 (47 x+37)}{2 \left (3 x^2+5 x+2\right )^2}+\frac{57210 x+47935}{6 x^2+10 x+4}+28917 \log (-6 x-4)-29125 \log (-2 (x+1))+208 \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-75*(37 + 47*x))/(2*(2 + 5*x + 3*x^2)^2) + (47935 + 57210*x)/(4 + 10*x + 6*x^2
) + 28917*Log[-4 - 6*x] - 29125*Log[-2*(1 + x)] + 208*Log[3 + 2*x])/125

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Maple [A]  time = 0.017, size = 56, normalized size = 0.8 \[ -{\frac{153}{10\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{2646}{50+75\,x}}+{\frac{28917\,\ln \left ( 2+3\,x \right ) }{125}}+{\frac{208\,\ln \left ( 3+2\,x \right ) }{125}}+3\, \left ( 1+x \right ) ^{-2}+41\, \left ( 1+x \right ) ^{-1}-233\,\ln \left ( 1+x \right ) \]

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]

-153/10/(2+3*x)^2+2646/25/(2+3*x)+28917/125*ln(2+3*x)+208/125*ln(3+2*x)+3/(1+x)^
2+41/(1+x)-233*ln(1+x)

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Maxima [A]  time = 0.689091, size = 84, normalized size = 1.22 \[ \frac{34326 \, x^{3} + 85971 \, x^{2} + 70114 \, x + 18619}{50 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + \frac{28917}{125} \, \log \left (3 \, x + 2\right ) + \frac{208}{125} \, \log \left (2 \, x + 3\right ) - 233 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/50*(34326*x^3 + 85971*x^2 + 70114*x + 18619)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x +
4) + 28917/125*log(3*x + 2) + 208/125*log(2*x + 3) - 233*log(x + 1)

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Fricas [A]  time = 0.267851, size = 163, normalized size = 2.36 \[ \frac{171630 \, x^{3} + 429855 \, x^{2} + 57834 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 416 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (2 \, x + 3\right ) - 58250 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) + 350570 \, x + 93095}{250 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/250*(171630*x^3 + 429855*x^2 + 57834*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(
3*x + 2) + 416*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(2*x + 3) - 58250*(9*x^4
+ 30*x^3 + 37*x^2 + 20*x + 4)*log(x + 1) + 350570*x + 93095)/(9*x^4 + 30*x^3 + 3
7*x^2 + 20*x + 4)

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Sympy [A]  time = 0.600716, size = 61, normalized size = 0.88 \[ \frac{34326 x^{3} + 85971 x^{2} + 70114 x + 18619}{450 x^{4} + 1500 x^{3} + 1850 x^{2} + 1000 x + 200} + \frac{28917 \log{\left (x + \frac{2}{3} \right )}}{125} - 233 \log{\left (x + 1 \right )} + \frac{208 \log{\left (x + \frac{3}{2} \right )}}{125} \]

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]

(34326*x**3 + 85971*x**2 + 70114*x + 18619)/(450*x**4 + 1500*x**3 + 1850*x**2 +
1000*x + 200) + 28917*log(x + 2/3)/125 - 233*log(x + 1) + 208*log(x + 3/2)/125

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GIAC/XCAS [A]  time = 0.269978, size = 74, normalized size = 1.07 \[ \frac{34326 \, x^{3} + 85971 \, x^{2} + 70114 \, x + 18619}{50 \,{\left (3 \, x + 2\right )}^{2}{\left (x + 1\right )}^{2}} + \frac{28917}{125} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{208}{125} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - 233 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/50*(34326*x^3 + 85971*x^2 + 70114*x + 18619)/((3*x + 2)^2*(x + 1)^2) + 28917/1
25*ln(abs(3*x + 2)) + 208/125*ln(abs(2*x + 3)) - 233*ln(abs(x + 1))