3.2392 \(\int \frac{5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=66 \[ -\frac{3 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )}-\frac{454}{25 (2 x+3)}+11 \log (x+1)+\frac{812}{125} \log (2 x+3)-\frac{2187}{125} \log (3 x+2) \]

[Out]

-454/(25*(3 + 2*x)) - (3*(37 + 47*x))/(5*(3 + 2*x)*(2 + 5*x + 3*x^2)) + 11*Log[1
+ x] + (812*Log[3 + 2*x])/125 - (2187*Log[2 + 3*x])/125

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Rubi [A]  time = 0.107741, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{3 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )}-\frac{454}{25 (2 x+3)}+11 \log (x+1)+\frac{812}{125} \log (2 x+3)-\frac{2187}{125} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

-454/(25*(3 + 2*x)) - (3*(37 + 47*x))/(5*(3 + 2*x)*(2 + 5*x + 3*x^2)) + 11*Log[1
+ x] + (812*Log[3 + 2*x])/125 - (2187*Log[2 + 3*x])/125

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Rubi in Sympy [A]  time = 21.0164, size = 54, normalized size = 0.82 \[ 11 \log{\left (x + 1 \right )} + \frac{812 \log{\left (2 x + 3 \right )}}{125} - \frac{2187 \log{\left (3 x + 2 \right )}}{125} - \frac{141 x + 111}{5 \left (2 x + 3\right ) \left (3 x^{2} + 5 x + 2\right )} - \frac{454}{25 \left (2 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11*log(x + 1) + 812*log(2*x + 3)/125 - 2187*log(3*x + 2)/125 - (141*x + 111)/(5*
(2*x + 3)*(3*x**2 + 5*x + 2)) - 454/(25*(2*x + 3))

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Mathematica [A]  time = 0.0515185, size = 57, normalized size = 0.86 \[ \frac{1}{125} \left (-\frac{15 (201 x+151)}{3 x^2+5 x+2}-\frac{260}{2 x+3}-2187 \log (-6 x-4)+1375 \log (-2 (x+1))+812 \log (2 x+3)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-260/(3 + 2*x) - (15*(151 + 201*x))/(2 + 5*x + 3*x^2) - 2187*Log[-4 - 6*x] + 13
75*Log[-2*(1 + x)] + 812*Log[3 + 2*x])/125

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Maple [A]  time = 0.019, size = 49, normalized size = 0.7 \[ -{\frac{153}{50+75\,x}}-{\frac{2187\,\ln \left ( 2+3\,x \right ) }{125}}-{\frac{52}{75+50\,x}}+{\frac{812\,\ln \left ( 3+2\,x \right ) }{125}}-6\, \left ( 1+x \right ) ^{-1}+11\,\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-153/25/(2+3*x)-2187/125*ln(2+3*x)-52/25/(3+2*x)+812/125*ln(3+2*x)-6/(1+x)+11*ln
(1+x)

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Maxima [A]  time = 0.689627, size = 70, normalized size = 1.06 \[ -\frac{1362 \, x^{2} + 2975 \, x + 1463}{25 \,{\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )}} - \frac{2187}{125} \, \log \left (3 \, x + 2\right ) + \frac{812}{125} \, \log \left (2 \, x + 3\right ) + 11 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/25*(1362*x^2 + 2975*x + 1463)/(6*x^3 + 19*x^2 + 19*x + 6) - 2187/125*log(3*x
+ 2) + 812/125*log(2*x + 3) + 11*log(x + 1)

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Fricas [A]  time = 0.266771, size = 130, normalized size = 1.97 \[ -\frac{6810 \, x^{2} + 2187 \,{\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )} \log \left (3 \, x + 2\right ) - 812 \,{\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )} \log \left (2 \, x + 3\right ) - 1375 \,{\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )} \log \left (x + 1\right ) + 14875 \, x + 7315}{125 \,{\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/125*(6810*x^2 + 2187*(6*x^3 + 19*x^2 + 19*x + 6)*log(3*x + 2) - 812*(6*x^3 +
19*x^2 + 19*x + 6)*log(2*x + 3) - 1375*(6*x^3 + 19*x^2 + 19*x + 6)*log(x + 1) +
14875*x + 7315)/(6*x^3 + 19*x^2 + 19*x + 6)

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Sympy [A]  time = 0.522059, size = 51, normalized size = 0.77 \[ - \frac{1362 x^{2} + 2975 x + 1463}{150 x^{3} + 475 x^{2} + 475 x + 150} - \frac{2187 \log{\left (x + \frac{2}{3} \right )}}{125} + 11 \log{\left (x + 1 \right )} + \frac{812 \log{\left (x + \frac{3}{2} \right )}}{125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(1362*x**2 + 2975*x + 1463)/(150*x**3 + 475*x**2 + 475*x + 150) - 2187*log(x +
2/3)/125 + 11*log(x + 1) + 812*log(x + 3/2)/125

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GIAC/XCAS [A]  time = 0.298274, size = 104, normalized size = 1.58 \[ -\frac{52}{25 \,{\left (2 \, x + 3\right )}} + \frac{6 \,{\left (\frac{1403}{2 \, x + 3} - 903\right )}}{125 \,{\left (\frac{5}{2 \, x + 3} - 3\right )}{\left (\frac{1}{2 \, x + 3} - 1\right )}} + 11 \,{\rm ln}\left ({\left | -\frac{1}{2 \, x + 3} + 1 \right |}\right ) - \frac{2187}{125} \,{\rm ln}\left ({\left | -\frac{5}{2 \, x + 3} + 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-52/25/(2*x + 3) + 6/125*(1403/(2*x + 3) - 903)/((5/(2*x + 3) - 3)*(1/(2*x + 3)
- 1)) + 11*ln(abs(-1/(2*x + 3) + 1)) - 2187/125*ln(abs(-5/(2*x + 3) + 3))