3.1429 \(\int \frac{3+5 x}{(1-2 x) (2+3 x)^4} \, dx\)

Optimal. Leaf size=54 \[ -\frac{22}{343 (3 x+2)}-\frac{11}{98 (3 x+2)^2}+\frac{1}{63 (3 x+2)^3}-\frac{44 \log (1-2 x)}{2401}+\frac{44 \log (3 x+2)}{2401} \]

[Out]

1/(63*(2 + 3*x)^3) - 11/(98*(2 + 3*x)^2) - 22/(343*(2 + 3*x)) - (44*Log[1 - 2*x]
)/2401 + (44*Log[2 + 3*x])/2401

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Rubi [A]  time = 0.0508309, 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 \[ -\frac{22}{343 (3 x+2)}-\frac{11}{98 (3 x+2)^2}+\frac{1}{63 (3 x+2)^3}-\frac{44 \log (1-2 x)}{2401}+\frac{44 \log (3 x+2)}{2401} \]

Antiderivative was successfully verified.

[In]  Int[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^4),x]

[Out]

1/(63*(2 + 3*x)^3) - 11/(98*(2 + 3*x)^2) - 22/(343*(2 + 3*x)) - (44*Log[1 - 2*x]
)/2401 + (44*Log[2 + 3*x])/2401

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Rubi in Sympy [A]  time = 8.44748, size = 46, normalized size = 0.85 \[ - \frac{44 \log{\left (- 2 x + 1 \right )}}{2401} + \frac{44 \log{\left (3 x + 2 \right )}}{2401} - \frac{22}{343 \left (3 x + 2\right )} - \frac{11}{98 \left (3 x + 2\right )^{2}} + \frac{1}{63 \left (3 x + 2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3+5*x)/(1-2*x)/(2+3*x)**4,x)

[Out]

-44*log(-2*x + 1)/2401 + 44*log(3*x + 2)/2401 - 22/(343*(3*x + 2)) - 11/(98*(3*x
+ 2)**2) + 1/(63*(3*x + 2)**3)

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Mathematica [A]  time = 0.0356704, size = 40, normalized size = 0.74 \[ \frac{-\frac{7 \left (3564 x^2+6831 x+2872\right )}{(3 x+2)^3}-792 \log (3-6 x)+792 \log (3 x+2)}{43218} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^4),x]

[Out]

((-7*(2872 + 6831*x + 3564*x^2))/(2 + 3*x)^3 - 792*Log[3 - 6*x] + 792*Log[2 + 3*
x])/43218

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Maple [A]  time = 0.011, size = 45, normalized size = 0.8 \[{\frac{1}{63\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{11}{98\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{22}{686+1029\,x}}+{\frac{44\,\ln \left ( 2+3\,x \right ) }{2401}}-{\frac{44\,\ln \left ( -1+2\,x \right ) }{2401}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)/(1-2*x)/(2+3*x)^4,x)

[Out]

1/63/(2+3*x)^3-11/98/(2+3*x)^2-22/343/(2+3*x)+44/2401*ln(2+3*x)-44/2401*ln(-1+2*
x)

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Maxima [A]  time = 1.35157, size = 62, normalized size = 1.15 \[ -\frac{3564 \, x^{2} + 6831 \, x + 2872}{6174 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{44}{2401} \, \log \left (3 \, x + 2\right ) - \frac{44}{2401} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x + 3)/((3*x + 2)^4*(2*x - 1)),x, algorithm="maxima")

[Out]

-1/6174*(3564*x^2 + 6831*x + 2872)/(27*x^3 + 54*x^2 + 36*x + 8) + 44/2401*log(3*
x + 2) - 44/2401*log(2*x - 1)

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Fricas [A]  time = 0.225705, size = 101, normalized size = 1.87 \[ -\frac{24948 \, x^{2} - 792 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 792 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (2 \, x - 1\right ) + 47817 \, x + 20104}{43218 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x + 3)/((3*x + 2)^4*(2*x - 1)),x, algorithm="fricas")

[Out]

-1/43218*(24948*x^2 - 792*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) + 792*(27*x^
3 + 54*x^2 + 36*x + 8)*log(2*x - 1) + 47817*x + 20104)/(27*x^3 + 54*x^2 + 36*x +
8)

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Sympy [A]  time = 0.388892, size = 44, normalized size = 0.81 \[ - \frac{3564 x^{2} + 6831 x + 2872}{166698 x^{3} + 333396 x^{2} + 222264 x + 49392} - \frac{44 \log{\left (x - \frac{1}{2} \right )}}{2401} + \frac{44 \log{\left (x + \frac{2}{3} \right )}}{2401} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3+5*x)/(1-2*x)/(2+3*x)**4,x)

[Out]

-(3564*x**2 + 6831*x + 2872)/(166698*x**3 + 333396*x**2 + 222264*x + 49392) - 44
*log(x - 1/2)/2401 + 44*log(x + 2/3)/2401

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GIAC/XCAS [A]  time = 0.208378, size = 51, normalized size = 0.94 \[ -\frac{3564 \, x^{2} + 6831 \, x + 2872}{6174 \,{\left (3 \, x + 2\right )}^{3}} + \frac{44}{2401} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - \frac{44}{2401} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x + 3)/((3*x + 2)^4*(2*x - 1)),x, algorithm="giac")

[Out]

-1/6174*(3564*x^2 + 6831*x + 2872)/(3*x + 2)^3 + 44/2401*ln(abs(3*x + 2)) - 44/2
401*ln(abs(2*x - 1))