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

Optimal. Leaf size=29 \[ -\frac{11}{14} \log (3-x)+\frac{3}{2} \log (5-x)+\frac{2}{7} \log (x+4) \]

[Out]

(-11*Log[3 - x])/14 + (3*Log[5 - x])/2 + (2*Log[4 + x])/7

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Rubi [A]  time = 0.0939585, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{11}{14} \log (3-x)+\frac{3}{2} \log (5-x)+\frac{2}{7} \log (x+4) \]

Antiderivative was successfully verified.

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

[Out]

(-11*Log[3 - x])/14 + (3*Log[5 - x])/2 + (2*Log[4 + x])/7

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.0118387, size = 29, normalized size = 1. \[ -\frac{11}{14} \log (3-x)+\frac{3}{2} \log (5-x)+\frac{2}{7} \log (x+4) \]

Antiderivative was successfully verified.

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

[Out]

(-11*Log[3 - x])/14 + (3*Log[5 - x])/2 + (2*Log[4 + x])/7

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Maple [A]  time = 0.011, size = 20, normalized size = 0.7 \[ -{\frac{11\,\ln \left ( -3+x \right ) }{14}}+{\frac{2\,\ln \left ( 4+x \right ) }{7}}+{\frac{3\,\ln \left ( -5+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11/14*ln(-3+x)+2/7*ln(4+x)+3/2*ln(-5+x)

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Maxima [A]  time = 0.815033, size = 26, normalized size = 0.9 \[ \frac{2}{7} \, \log \left (x + 4\right ) - \frac{11}{14} \, \log \left (x - 3\right ) + \frac{3}{2} \, \log \left (x - 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*log(x + 4) - 11/14*log(x - 3) + 3/2*log(x - 5)

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Fricas [A]  time = 0.256919, size = 26, normalized size = 0.9 \[ \frac{2}{7} \, \log \left (x + 4\right ) - \frac{11}{14} \, \log \left (x - 3\right ) + \frac{3}{2} \, \log \left (x - 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*log(x + 4) - 11/14*log(x - 3) + 3/2*log(x - 5)

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Sympy [A]  time = 0.309087, size = 24, normalized size = 0.83 \[ \frac{3 \log{\left (x - 5 \right )}}{2} - \frac{11 \log{\left (x - 3 \right )}}{14} + \frac{2 \log{\left (x + 4 \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(x - 5)/2 - 11*log(x - 3)/14 + 2*log(x + 4)/7

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GIAC/XCAS [A]  time = 0.258399, size = 30, normalized size = 1.03 \[ \frac{2}{7} \,{\rm ln}\left ({\left | x + 4 \right |}\right ) - \frac{11}{14} \,{\rm ln}\left ({\left | x - 3 \right |}\right ) + \frac{3}{2} \,{\rm ln}\left ({\left | x - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*ln(abs(x + 4)) - 11/14*ln(abs(x - 3)) + 3/2*ln(abs(x - 5))