3.899 \(\int \frac{7+3 x}{8+6 x+x^2} \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{2} \log (x+2)+\frac{5}{2} \log (x+4) \]

[Out]

Log[2 + x]/2 + (5*Log[4 + x])/2

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Rubi [A]  time = 0.0146914, antiderivative size = 17, 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 \[ \frac{1}{2} \log (x+2)+\frac{5}{2} \log (x+4) \]

Antiderivative was successfully verified.

[In]  Int[(7 + 3*x)/(8 + 6*x + x^2),x]

[Out]

Log[2 + x]/2 + (5*Log[4 + x])/2

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Rubi in Sympy [A]  time = 4.33713, size = 14, normalized size = 0.82 \[ \frac{\log{\left (x + 2 \right )}}{2} + \frac{5 \log{\left (x + 4 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((7+3*x)/(x**2+6*x+8),x)

[Out]

log(x + 2)/2 + 5*log(x + 4)/2

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Mathematica [A]  time = 0.00747192, size = 17, normalized size = 1. \[ \frac{1}{2} \log (x+2)+\frac{5}{2} \log (x+4) \]

Antiderivative was successfully verified.

[In]  Integrate[(7 + 3*x)/(8 + 6*x + x^2),x]

[Out]

Log[2 + x]/2 + (5*Log[4 + x])/2

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Maple [A]  time = 0.008, size = 14, normalized size = 0.8 \[{\frac{\ln \left ( 2+x \right ) }{2}}+{\frac{5\,\ln \left ( 4+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((7+3*x)/(x^2+6*x+8),x)

[Out]

1/2*ln(2+x)+5/2*ln(4+x)

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Maxima [A]  time = 0.691366, size = 18, normalized size = 1.06 \[ \frac{5}{2} \, \log \left (x + 4\right ) + \frac{1}{2} \, \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 7)/(x^2 + 6*x + 8),x, algorithm="maxima")

[Out]

5/2*log(x + 4) + 1/2*log(x + 2)

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Fricas [A]  time = 0.277126, size = 18, normalized size = 1.06 \[ \frac{5}{2} \, \log \left (x + 4\right ) + \frac{1}{2} \, \log \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 7)/(x^2 + 6*x + 8),x, algorithm="fricas")

[Out]

5/2*log(x + 4) + 1/2*log(x + 2)

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Sympy [A]  time = 0.193692, size = 14, normalized size = 0.82 \[ \frac{\log{\left (x + 2 \right )}}{2} + \frac{5 \log{\left (x + 4 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((7+3*x)/(x**2+6*x+8),x)

[Out]

log(x + 2)/2 + 5*log(x + 4)/2

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GIAC/XCAS [A]  time = 0.267127, size = 20, normalized size = 1.18 \[ \frac{5}{2} \,{\rm ln}\left ({\left | x + 4 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x + 7)/(x^2 + 6*x + 8),x, algorithm="giac")

[Out]

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