3.1427 \(\int \frac{3+5 x}{(1-2 x) (2+3 x)^2} \, dx\)

Optimal. Leaf size=32 \[ \frac{1}{21 (3 x+2)}-\frac{11}{49} \log (1-2 x)+\frac{11}{49} \log (3 x+2) \]

[Out]

1/(21*(2 + 3*x)) - (11*Log[1 - 2*x])/49 + (11*Log[2 + 3*x])/49

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Rubi [A]  time = 0.0409236, antiderivative size = 32, 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 \[ \frac{1}{21 (3 x+2)}-\frac{11}{49} \log (1-2 x)+\frac{11}{49} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

1/(21*(2 + 3*x)) - (11*Log[1 - 2*x])/49 + (11*Log[2 + 3*x])/49

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Rubi in Sympy [A]  time = 6.34361, size = 26, normalized size = 0.81 \[ - \frac{11 \log{\left (- 2 x + 1 \right )}}{49} + \frac{11 \log{\left (3 x + 2 \right )}}{49} + \frac{1}{21 \left (3 x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11*log(-2*x + 1)/49 + 11*log(3*x + 2)/49 + 1/(21*(3*x + 2))

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Mathematica [A]  time = 0.0241222, size = 30, normalized size = 0.94 \[ \frac{1}{147} \left (\frac{7}{3 x+2}-33 \log (3-6 x)+33 \log (3 x+2)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(7/(2 + 3*x) - 33*Log[3 - 6*x] + 33*Log[2 + 3*x])/147

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Maple [A]  time = 0.012, size = 27, normalized size = 0.8 \[{\frac{1}{42+63\,x}}+{\frac{11\,\ln \left ( 2+3\,x \right ) }{49}}-{\frac{11\,\ln \left ( -1+2\,x \right ) }{49}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21/(2+3*x)+11/49*ln(2+3*x)-11/49*ln(-1+2*x)

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Maxima [A]  time = 1.34336, size = 35, normalized size = 1.09 \[ \frac{1}{21 \,{\left (3 \, x + 2\right )}} + \frac{11}{49} \, \log \left (3 \, x + 2\right ) - \frac{11}{49} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21/(3*x + 2) + 11/49*log(3*x + 2) - 11/49*log(2*x - 1)

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Fricas [A]  time = 0.211702, size = 50, normalized size = 1.56 \[ \frac{33 \,{\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 33 \,{\left (3 \, x + 2\right )} \log \left (2 \, x - 1\right ) + 7}{147 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/147*(33*(3*x + 2)*log(3*x + 2) - 33*(3*x + 2)*log(2*x - 1) + 7)/(3*x + 2)

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Sympy [A]  time = 0.285818, size = 26, normalized size = 0.81 \[ - \frac{11 \log{\left (x - \frac{1}{2} \right )}}{49} + \frac{11 \log{\left (x + \frac{2}{3} \right )}}{49} + \frac{1}{63 x + 42} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11*log(x - 1/2)/49 + 11*log(x + 2/3)/49 + 1/(63*x + 42)

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GIAC/XCAS [A]  time = 0.21094, size = 34, normalized size = 1.06 \[ \frac{1}{21 \,{\left (3 \, x + 2\right )}} - \frac{11}{49} \,{\rm ln}\left ({\left | -\frac{7}{3 \, x + 2} + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21/(3*x + 2) - 11/49*ln(abs(-7/(3*x + 2) + 2))