3.189 \(\int \frac{x^2}{(-3+x) (2+x)^2} \, dx\)

Optimal. Leaf size=28 \[ \frac{4}{5 (x+2)}+\frac{9}{25} \log (3-x)+\frac{16}{25} \log (x+2) \]

[Out]

4/(5*(2 + x)) + (9*Log[3 - x])/25 + (16*Log[2 + x])/25

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Rubi [A]  time = 0.0325676, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{4}{5 (x+2)}+\frac{9}{25} \log (3-x)+\frac{16}{25} \log (x+2) \]

Antiderivative was successfully verified.

[In]  Int[x^2/((-3 + x)*(2 + x)^2),x]

[Out]

4/(5*(2 + x)) + (9*Log[3 - x])/25 + (16*Log[2 + x])/25

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Rubi in Sympy [A]  time = 2.05015, size = 22, normalized size = 0.79 \[ \frac{9 \log{\left (- x + 3 \right )}}{25} + \frac{16 \log{\left (x + 2 \right )}}{25} + \frac{4}{5 \left (x + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*log(-x + 3)/25 + 16*log(x + 2)/25 + 4/(5*(x + 2))

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Mathematica [A]  time = 0.0235232, size = 26, normalized size = 0.93 \[ \frac{4}{5 (x+2)}+\frac{9}{25} \log (x-3)+\frac{16}{25} \log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

4/(5*(2 + x)) + (9*Log[-3 + x])/25 + (16*Log[2 + x])/25

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Maple [A]  time = 0.01, size = 21, normalized size = 0.8 \[{\frac{9\,\ln \left ( -3+x \right ) }{25}}+{\frac{4}{10+5\,x}}+{\frac{16\,\ln \left ( 2+x \right ) }{25}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-3+x)/(2+x)^2,x)

[Out]

9/25*ln(-3+x)+4/5/(2+x)+16/25*ln(2+x)

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Maxima [A]  time = 1.36354, size = 27, normalized size = 0.96 \[ \frac{4}{5 \,{\left (x + 2\right )}} + \frac{16}{25} \, \log \left (x + 2\right ) + \frac{9}{25} \, \log \left (x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/5/(x + 2) + 16/25*log(x + 2) + 9/25*log(x - 3)

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Fricas [A]  time = 0.203826, size = 36, normalized size = 1.29 \[ \frac{16 \,{\left (x + 2\right )} \log \left (x + 2\right ) + 9 \,{\left (x + 2\right )} \log \left (x - 3\right ) + 20}{25 \,{\left (x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25*(16*(x + 2)*log(x + 2) + 9*(x + 2)*log(x - 3) + 20)/(x + 2)

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Sympy [A]  time = 0.129302, size = 22, normalized size = 0.79 \[ \frac{9 \log{\left (x - 3 \right )}}{25} + \frac{16 \log{\left (x + 2 \right )}}{25} + \frac{4}{5 x + 10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*log(x - 3)/25 + 16*log(x + 2)/25 + 4/(5*x + 10)

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GIAC/XCAS [A]  time = 0.206013, size = 35, normalized size = 1.25 \[ \frac{4}{5 \,{\left (x + 2\right )}} +{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{9}{25} \,{\rm ln}\left ({\left | -\frac{5}{x + 2} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/5/(x + 2) + ln(abs(x + 2)) + 9/25*ln(abs(-5/(x + 2) + 1))