3.1136 \(\int \frac{(1-2 x) (3+5 x)}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=27 \[ -\frac{10 x}{9}+\frac{7}{27 (3 x+2)}+\frac{37}{27} \log (3 x+2) \]

[Out]

(-10*x)/9 + 7/(27*(2 + 3*x)) + (37*Log[2 + 3*x])/27

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Rubi [A]  time = 0.0356055, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{10 x}{9}+\frac{7}{27 (3 x+2)}+\frac{37}{27} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

(-10*x)/9 + 7/(27*(2 + 3*x)) + (37*Log[2 + 3*x])/27

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{37 \log{\left (3 x + 2 \right )}}{27} + \int \left (- \frac{10}{9}\right )\, dx + \frac{7}{27 \left (3 x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

37*log(3*x + 2)/27 + Integral(-10/9, x) + 7/(27*(3*x + 2))

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Mathematica [A]  time = 0.0166986, size = 26, normalized size = 0.96 \[ \frac{1}{27} \left (-30 x+\frac{7}{3 x+2}+37 \log (3 x+2)-20\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-20 - 30*x + 7/(2 + 3*x) + 37*Log[2 + 3*x])/27

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Maple [A]  time = 0.009, size = 22, normalized size = 0.8 \[ -{\frac{10\,x}{9}}+{\frac{7}{54+81\,x}}+{\frac{37\,\ln \left ( 2+3\,x \right ) }{27}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10/9*x+7/27/(2+3*x)+37/27*ln(2+3*x)

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Maxima [A]  time = 1.34248, size = 28, normalized size = 1.04 \[ -\frac{10}{9} \, x + \frac{7}{27 \,{\left (3 \, x + 2\right )}} + \frac{37}{27} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10/9*x + 7/27/(3*x + 2) + 37/27*log(3*x + 2)

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Fricas [A]  time = 0.212003, size = 43, normalized size = 1.59 \[ -\frac{90 \, x^{2} - 37 \,{\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) + 60 \, x - 7}{27 \,{\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(90*x^2 - 37*(3*x + 2)*log(3*x + 2) + 60*x - 7)/(3*x + 2)

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Sympy [A]  time = 0.212192, size = 20, normalized size = 0.74 \[ - \frac{10 x}{9} + \frac{37 \log{\left (3 x + 2 \right )}}{27} + \frac{7}{81 x + 54} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*x/9 + 37*log(3*x + 2)/27 + 7/(81*x + 54)

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GIAC/XCAS [A]  time = 0.215001, size = 43, normalized size = 1.59 \[ -\frac{10}{9} \, x + \frac{7}{27 \,{\left (3 \, x + 2\right )}} - \frac{37}{27} \,{\rm ln}\left (\frac{{\left | 3 \, x + 2 \right |}}{3 \,{\left (3 \, x + 2\right )}^{2}}\right ) - \frac{20}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10/9*x + 7/27/(3*x + 2) - 37/27*ln(1/3*abs(3*x + 2)/(3*x + 2)^2) - 20/27