3.1428 \(\int \frac{3+5 x}{(1-2 x) (2+3 x)^3} \, dx\)

Optimal. Leaf size=43 \[ -\frac{11}{49 (3 x+2)}+\frac{1}{42 (3 x+2)^2}-\frac{22}{343} \log (1-2 x)+\frac{22}{343} \log (3 x+2) \]

[Out]

1/(42*(2 + 3*x)^2) - 11/(49*(2 + 3*x)) - (22*Log[1 - 2*x])/343 + (22*Log[2 + 3*x
])/343

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Rubi [A]  time = 0.044414, antiderivative size = 43, 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 \[ -\frac{11}{49 (3 x+2)}+\frac{1}{42 (3 x+2)^2}-\frac{22}{343} \log (1-2 x)+\frac{22}{343} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

1/(42*(2 + 3*x)^2) - 11/(49*(2 + 3*x)) - (22*Log[1 - 2*x])/343 + (22*Log[2 + 3*x
])/343

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Rubi in Sympy [A]  time = 7.37269, size = 36, normalized size = 0.84 \[ - \frac{22 \log{\left (- 2 x + 1 \right )}}{343} + \frac{22 \log{\left (3 x + 2 \right )}}{343} - \frac{11}{49 \left (3 x + 2\right )} + \frac{1}{42 \left (3 x + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-22*log(-2*x + 1)/343 + 22*log(3*x + 2)/343 - 11/(49*(3*x + 2)) + 1/(42*(3*x + 2
)**2)

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Mathematica [A]  time = 0.0294692, size = 35, normalized size = 0.81 \[ \frac{-\frac{7 (198 x+125)}{(3 x+2)^2}-132 \log (3-6 x)+132 \log (3 x+2)}{2058} \]

Antiderivative was successfully verified.

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

[Out]

((-7*(125 + 198*x))/(2 + 3*x)^2 - 132*Log[3 - 6*x] + 132*Log[2 + 3*x])/2058

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Maple [A]  time = 0.011, size = 36, normalized size = 0.8 \[{\frac{1}{42\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{11}{98+147\,x}}+{\frac{22\,\ln \left ( 2+3\,x \right ) }{343}}-{\frac{22\,\ln \left ( -1+2\,x \right ) }{343}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/42/(2+3*x)^2-11/49/(2+3*x)+22/343*ln(2+3*x)-22/343*ln(-1+2*x)

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Maxima [A]  time = 1.34715, size = 49, normalized size = 1.14 \[ -\frac{198 \, x + 125}{294 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{22}{343} \, \log \left (3 \, x + 2\right ) - \frac{22}{343} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/294*(198*x + 125)/(9*x^2 + 12*x + 4) + 22/343*log(3*x + 2) - 22/343*log(2*x -
1)

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Fricas [A]  time = 0.210519, size = 74, normalized size = 1.72 \[ \frac{132 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 132 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (2 \, x - 1\right ) - 1386 \, x - 875}{2058 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2058*(132*(9*x^2 + 12*x + 4)*log(3*x + 2) - 132*(9*x^2 + 12*x + 4)*log(2*x - 1
) - 1386*x - 875)/(9*x^2 + 12*x + 4)

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Sympy [A]  time = 0.338226, size = 34, normalized size = 0.79 \[ - \frac{198 x + 125}{2646 x^{2} + 3528 x + 1176} - \frac{22 \log{\left (x - \frac{1}{2} \right )}}{343} + \frac{22 \log{\left (x + \frac{2}{3} \right )}}{343} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(198*x + 125)/(2646*x**2 + 3528*x + 1176) - 22*log(x - 1/2)/343 + 22*log(x + 2/
3)/343

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GIAC/XCAS [A]  time = 0.206726, size = 45, normalized size = 1.05 \[ -\frac{198 \, x + 125}{294 \,{\left (3 \, x + 2\right )}^{2}} + \frac{22}{343} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) - \frac{22}{343} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/294*(198*x + 125)/(3*x + 2)^2 + 22/343*ln(abs(3*x + 2)) - 22/343*ln(abs(2*x -
1))