3.153 \(\int \frac{5+x}{-2+x+x^2} \, dx\)

Optimal. Leaf size=15 \[ 2 \log (1-x)-\log (x+2) \]

[Out]

2*Log[1 - x] - Log[2 + x]

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Rubi [A]  time = 0.0126646, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ 2 \log (1-x)-\log (x+2) \]

Antiderivative was successfully verified.

[In]  Int[(5 + x)/(-2 + x + x^2),x]

[Out]

2*Log[1 - x] - Log[2 + x]

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Rubi in Sympy [A]  time = 1.67623, size = 10, normalized size = 0.67 \[ 2 \log{\left (- x + 1 \right )} - \log{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(-x + 1) - log(x + 2)

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Mathematica [A]  time = 0.00568866, size = 15, normalized size = 1. \[ 2 \log (1-x)-\log (x+2) \]

Antiderivative was successfully verified.

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

[Out]

2*Log[1 - x] - Log[2 + x]

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Maple [A]  time = 0.007, size = 14, normalized size = 0.9 \[ -\ln \left ( 2+x \right ) +2\,\ln \left ( -1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5+x)/(x^2+x-2),x)

[Out]

-ln(2+x)+2*ln(-1+x)

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Maxima [A]  time = 1.47329, size = 18, normalized size = 1.2 \[ -\log \left (x + 2\right ) + 2 \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x + 2) + 2*log(x - 1)

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Fricas [A]  time = 0.19868, size = 18, normalized size = 1.2 \[ -\log \left (x + 2\right ) + 2 \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x + 2) + 2*log(x - 1)

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Sympy [A]  time = 0.091234, size = 10, normalized size = 0.67 \[ 2 \log{\left (x - 1 \right )} - \log{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5+x)/(x**2+x-2),x)

[Out]

2*log(x - 1) - log(x + 2)

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GIAC/XCAS [A]  time = 0.207374, size = 20, normalized size = 1.33 \[ -{\rm ln}\left ({\left | x + 2 \right |}\right ) + 2 \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(abs(x + 2)) + 2*ln(abs(x - 1))