3.1623 \(\int \frac{3+5 x}{(1-2 x)^3 (2+3 x)^2} \, dx\)

Optimal. Leaf size=54 \[ \frac{31}{343 (1-2 x)}+\frac{3}{343 (3 x+2)}+\frac{11}{98 (1-2 x)^2}-\frac{87 \log (1-2 x)}{2401}+\frac{87 \log (3 x+2)}{2401} \]

[Out]

11/(98*(1 - 2*x)^2) + 31/(343*(1 - 2*x)) + 3/(343*(2 + 3*x)) - (87*Log[1 - 2*x])
/2401 + (87*Log[2 + 3*x])/2401

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Rubi [A]  time = 0.0590125, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{31}{343 (1-2 x)}+\frac{3}{343 (3 x+2)}+\frac{11}{98 (1-2 x)^2}-\frac{87 \log (1-2 x)}{2401}+\frac{87 \log (3 x+2)}{2401} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 5*x)/((1 - 2*x)^3*(2 + 3*x)^2),x]

[Out]

11/(98*(1 - 2*x)^2) + 31/(343*(1 - 2*x)) + 3/(343*(2 + 3*x)) - (87*Log[1 - 2*x])
/2401 + (87*Log[2 + 3*x])/2401

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Rubi in Sympy [A]  time = 8.49885, size = 42, normalized size = 0.78 \[ - \frac{87 \log{\left (- 2 x + 1 \right )}}{2401} + \frac{87 \log{\left (3 x + 2 \right )}}{2401} + \frac{3}{343 \left (3 x + 2\right )} + \frac{31}{343 \left (- 2 x + 1\right )} + \frac{11}{98 \left (- 2 x + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)/(1-2*x)**3/(2+3*x)**2,x)

[Out]

-87*log(-2*x + 1)/2401 + 87*log(3*x + 2)/2401 + 3/(343*(3*x + 2)) + 31/(343*(-2*
x + 1)) + 11/(98*(-2*x + 1)**2)

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Mathematica [A]  time = 0.0593812, size = 47, normalized size = 0.87 \[ \frac{\frac{7 \left (-348 x^2+145 x+284\right )}{(1-2 x)^2 (3 x+2)}-174 \log (1-2 x)+174 \log (6 x+4)}{4802} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 5*x)/((1 - 2*x)^3*(2 + 3*x)^2),x]

[Out]

((7*(284 + 145*x - 348*x^2))/((1 - 2*x)^2*(2 + 3*x)) - 174*Log[1 - 2*x] + 174*Lo
g[4 + 6*x])/4802

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Maple [A]  time = 0.014, size = 45, normalized size = 0.8 \[{\frac{3}{686+1029\,x}}+{\frac{87\,\ln \left ( 2+3\,x \right ) }{2401}}+{\frac{11}{98\, \left ( -1+2\,x \right ) ^{2}}}-{\frac{31}{-343+686\,x}}-{\frac{87\,\ln \left ( -1+2\,x \right ) }{2401}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)/(1-2*x)^3/(2+3*x)^2,x)

[Out]

3/343/(2+3*x)+87/2401*ln(2+3*x)+11/98/(-1+2*x)^2-31/343/(-1+2*x)-87/2401*ln(-1+2
*x)

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Maxima [A]  time = 1.32311, size = 62, normalized size = 1.15 \[ -\frac{348 \, x^{2} - 145 \, x - 284}{686 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} + \frac{87}{2401} \, \log \left (3 \, x + 2\right ) - \frac{87}{2401} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x + 3)/((3*x + 2)^2*(2*x - 1)^3),x, algorithm="maxima")

[Out]

-1/686*(348*x^2 - 145*x - 284)/(12*x^3 - 4*x^2 - 5*x + 2) + 87/2401*log(3*x + 2)
- 87/2401*log(2*x - 1)

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Fricas [A]  time = 0.208251, size = 101, normalized size = 1.87 \[ -\frac{2436 \, x^{2} - 174 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) + 174 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \log \left (2 \, x - 1\right ) - 1015 \, x - 1988}{4802 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x + 3)/((3*x + 2)^2*(2*x - 1)^3),x, algorithm="fricas")

[Out]

-1/4802*(2436*x^2 - 174*(12*x^3 - 4*x^2 - 5*x + 2)*log(3*x + 2) + 174*(12*x^3 -
4*x^2 - 5*x + 2)*log(2*x - 1) - 1015*x - 1988)/(12*x^3 - 4*x^2 - 5*x + 2)

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Sympy [A]  time = 0.378826, size = 44, normalized size = 0.81 \[ - \frac{348 x^{2} - 145 x - 284}{8232 x^{3} - 2744 x^{2} - 3430 x + 1372} - \frac{87 \log{\left (x - \frac{1}{2} \right )}}{2401} + \frac{87 \log{\left (x + \frac{2}{3} \right )}}{2401} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3+5*x)/(1-2*x)**3/(2+3*x)**2,x)

[Out]

-(348*x**2 - 145*x - 284)/(8232*x**3 - 2744*x**2 - 3430*x + 1372) - 87*log(x - 1
/2)/2401 + 87*log(x + 2/3)/2401

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GIAC/XCAS [A]  time = 0.208253, size = 69, normalized size = 1.28 \[ \frac{3}{343 \,{\left (3 \, x + 2\right )}} + \frac{6 \,{\left (\frac{448}{3 \, x + 2} - 95\right )}}{2401 \,{\left (\frac{7}{3 \, x + 2} - 2\right )}^{2}} - \frac{87}{2401} \,{\rm ln}\left ({\left | -\frac{7}{3 \, x + 2} + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x + 3)/((3*x + 2)^2*(2*x - 1)^3),x, algorithm="giac")

[Out]

3/343/(3*x + 2) + 6/2401*(448/(3*x + 2) - 95)/(7/(3*x + 2) - 2)^2 - 87/2401*ln(a
bs(-7/(3*x + 2) + 2))