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

Optimal. Leaf size=15 \[ 2 \log (1-x)+3 \log (x+3) \]

[Out]

2*Log[1 - x] + 3*Log[3 + x]

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Rubi [A]  time = 0.0148709, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ 2 \log (1-x)+3 \log (x+3) \]

Antiderivative was successfully verified.

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

[Out]

2*Log[1 - x] + 3*Log[3 + x]

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Rubi in Sympy [A]  time = 2.01697, size = 12, normalized size = 0.8 \[ 2 \log{\left (- x + 1 \right )} + 3 \log{\left (x + 3 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(-x + 1) + 3*log(x + 3)

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Mathematica [A]  time = 0.00618239, size = 15, normalized size = 1. \[ 2 \log (1-x)+3 \log (x+3) \]

Antiderivative was successfully verified.

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

[Out]

2*Log[1 - x] + 3*Log[3 + x]

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Maple [A]  time = 0.007, size = 14, normalized size = 0.9 \[ 2\,\ln \left ( -1+x \right ) +3\,\ln \left ( 3+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3+5*x)/(x^2+2*x-3),x)

[Out]

2*ln(-1+x)+3*ln(3+x)

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Maxima [A]  time = 1.37797, size = 18, normalized size = 1.2 \[ 3 \, \log \left (x + 3\right ) + 2 \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(x + 3) + 2*log(x - 1)

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Fricas [A]  time = 0.197893, size = 18, normalized size = 1.2 \[ 3 \, \log \left (x + 3\right ) + 2 \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(x + 3) + 2*log(x - 1)

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Sympy [A]  time = 0.114409, size = 12, normalized size = 0.8 \[ 2 \log{\left (x - 1 \right )} + 3 \log{\left (x + 3 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(x - 1) + 3*log(x + 3)

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GIAC/XCAS [A]  time = 0.221742, size = 20, normalized size = 1.33 \[ 3 \,{\rm ln}\left ({\left | x + 3 \right |}\right ) + 2 \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*ln(abs(x + 3)) + 2*ln(abs(x - 1))