3.2394 \(\int \frac{5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=88 \[ -\frac{3 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )}-\frac{16522}{625 (2 x+3)}-\frac{2212}{125 (2 x+3)^2}-\frac{1258}{75 (2 x+3)^3}-13 \log (x+1)+\frac{65816 \log (2 x+3)}{3125}-\frac{25191 \log (3 x+2)}{3125} \]

[Out]

-1258/(75*(3 + 2*x)^3) - 2212/(125*(3 + 2*x)^2) - 16522/(625*(3 + 2*x)) - (3*(37
+ 47*x))/(5*(3 + 2*x)^3*(2 + 5*x + 3*x^2)) - 13*Log[1 + x] + (65816*Log[3 + 2*x
])/3125 - (25191*Log[2 + 3*x])/3125

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Rubi [A]  time = 0.125687, antiderivative size = 88, 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 \[ -\frac{3 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )}-\frac{16522}{625 (2 x+3)}-\frac{2212}{125 (2 x+3)^2}-\frac{1258}{75 (2 x+3)^3}-13 \log (x+1)+\frac{65816 \log (2 x+3)}{3125}-\frac{25191 \log (3 x+2)}{3125} \]

Antiderivative was successfully verified.

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

[Out]

-1258/(75*(3 + 2*x)^3) - 2212/(125*(3 + 2*x)^2) - 16522/(625*(3 + 2*x)) - (3*(37
+ 47*x))/(5*(3 + 2*x)^3*(2 + 5*x + 3*x^2)) - 13*Log[1 + x] + (65816*Log[3 + 2*x
])/3125 - (25191*Log[2 + 3*x])/3125

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Rubi in Sympy [A]  time = 23.5371, size = 76, normalized size = 0.86 \[ - 13 \log{\left (x + 1 \right )} + \frac{65816 \log{\left (2 x + 3 \right )}}{3125} - \frac{25191 \log{\left (3 x + 2 \right )}}{3125} - \frac{16522}{625 \left (2 x + 3\right )} - \frac{2212}{125 \left (2 x + 3\right )^{2}} - \frac{141 x + 111}{5 \left (2 x + 3\right )^{3} \left (3 x^{2} + 5 x + 2\right )} - \frac{1258}{75 \left (2 x + 3\right )^{3}} \]

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]

-13*log(x + 1) + 65816*log(2*x + 3)/3125 - 25191*log(3*x + 2)/3125 - 16522/(625*
(2*x + 3)) - 2212/(125*(2*x + 3)**2) - (141*x + 111)/(5*(2*x + 3)**3*(3*x**2 + 5
*x + 2)) - 1258/(75*(2*x + 3)**3)

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Mathematica [A]  time = 0.0746357, size = 75, normalized size = 0.85 \[ \frac{-\frac{45 (4209 x+2959)}{3 x^2+5 x+2}-\frac{121560}{2 x+3}-\frac{30450}{(2 x+3)^2}-\frac{6500}{(2 x+3)^3}-75573 \log (-6 x-4)-121875 \log (-2 (x+1))+197448 \log (2 x+3)}{9375} \]

Antiderivative was successfully verified.

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

[Out]

(-6500/(3 + 2*x)^3 - 30450/(3 + 2*x)^2 - 121560/(3 + 2*x) - (45*(2959 + 4209*x))
/(2 + 5*x + 3*x^2) - 75573*Log[-4 - 6*x] - 121875*Log[-2*(1 + x)] + 197448*Log[3
+ 2*x])/9375

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Maple [A]  time = 0.02, size = 67, normalized size = 0.8 \[ -{\frac{1377}{1250+1875\,x}}-{\frac{25191\,\ln \left ( 2+3\,x \right ) }{3125}}-{\frac{52}{75\, \left ( 3+2\,x \right ) ^{3}}}-{\frac{406}{125\, \left ( 3+2\,x \right ) ^{2}}}-{\frac{8104}{1875+1250\,x}}+{\frac{65816\,\ln \left ( 3+2\,x \right ) }{3125}}-6\, \left ( 1+x \right ) ^{-1}-13\,\ln \left ( 1+x \right ) \]

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]

-1377/625/(2+3*x)-25191/3125*ln(2+3*x)-52/75/(3+2*x)^3-406/125/(3+2*x)^2-8104/62
5/(3+2*x)+65816/3125*ln(3+2*x)-6/(1+x)-13*ln(1+x)

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Maxima [A]  time = 0.691849, size = 97, normalized size = 1.1 \[ -\frac{594792 \, x^{4} + 2974776 \, x^{3} + 5433540 \, x^{2} + 4260599 \, x + 1195793}{1875 \,{\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )}} - \frac{25191}{3125} \, \log \left (3 \, x + 2\right ) + \frac{65816}{3125} \, \log \left (2 \, x + 3\right ) - 13 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1875*(594792*x^4 + 2974776*x^3 + 5433540*x^2 + 4260599*x + 1195793)/(24*x^5 +
148*x^4 + 358*x^3 + 423*x^2 + 243*x + 54) - 25191/3125*log(3*x + 2) + 65816/312
5*log(2*x + 3) - 13*log(x + 1)

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Fricas [A]  time = 0.267656, size = 197, normalized size = 2.24 \[ -\frac{2973960 \, x^{4} + 14873880 \, x^{3} + 27167700 \, x^{2} + 75573 \,{\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} \log \left (3 \, x + 2\right ) - 197448 \,{\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} \log \left (2 \, x + 3\right ) + 121875 \,{\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} \log \left (x + 1\right ) + 21302995 \, x + 5978965}{9375 \,{\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9375*(2973960*x^4 + 14873880*x^3 + 27167700*x^2 + 75573*(24*x^5 + 148*x^4 + 3
58*x^3 + 423*x^2 + 243*x + 54)*log(3*x + 2) - 197448*(24*x^5 + 148*x^4 + 358*x^3
+ 423*x^2 + 243*x + 54)*log(2*x + 3) + 121875*(24*x^5 + 148*x^4 + 358*x^3 + 423
*x^2 + 243*x + 54)*log(x + 1) + 21302995*x + 5978965)/(24*x^5 + 148*x^4 + 358*x^
3 + 423*x^2 + 243*x + 54)

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Sympy [A]  time = 0.669745, size = 71, normalized size = 0.81 \[ - \frac{594792 x^{4} + 2974776 x^{3} + 5433540 x^{2} + 4260599 x + 1195793}{45000 x^{5} + 277500 x^{4} + 671250 x^{3} + 793125 x^{2} + 455625 x + 101250} - \frac{25191 \log{\left (x + \frac{2}{3} \right )}}{3125} - 13 \log{\left (x + 1 \right )} + \frac{65816 \log{\left (x + \frac{3}{2} \right )}}{3125} \]

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]

-(594792*x**4 + 2974776*x**3 + 5433540*x**2 + 4260599*x + 1195793)/(45000*x**5 +
277500*x**4 + 671250*x**3 + 793125*x**2 + 455625*x + 101250) - 25191*log(x + 2/
3)/3125 - 13*log(x + 1) + 65816*log(x + 3/2)/3125

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GIAC/XCAS [A]  time = 0.267297, size = 90, normalized size = 1.02 \[ -\frac{594792 \, x^{4} + 2974776 \, x^{3} + 5433540 \, x^{2} + 4260599 \, x + 1195793}{1875 \,{\left (3 \, x + 2\right )}{\left (2 \, x + 3\right )}^{3}{\left (x + 1\right )}} - \frac{25191}{3125} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) + \frac{65816}{3125} \,{\rm ln}\left ({\left | 2 \, x + 3 \right |}\right ) - 13 \,{\rm ln}\left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1875*(594792*x^4 + 2974776*x^3 + 5433540*x^2 + 4260599*x + 1195793)/((3*x + 2
)*(2*x + 3)^3*(x + 1)) - 25191/3125*ln(abs(3*x + 2)) + 65816/3125*ln(abs(2*x + 3
)) - 13*ln(abs(x + 1))