3.900 \(\int \frac{5+2 x}{5+4 x+x^2} \, dx\)

Optimal. Leaf size=14 \[ \log \left (x^2+4 x+5\right )+\tan ^{-1}(x+2) \]

[Out]

ArcTan[2 + x] + Log[5 + 4*x + x^2]

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Rubi [A]  time = 0.0225822, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \log \left (x^2+4 x+5\right )+\tan ^{-1}(x+2) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[2 + x] + Log[5 + 4*x + x^2]

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Rubi in Sympy [A]  time = 5.68422, size = 14, normalized size = 1. \[ \log{\left (x^{2} + 4 x + 5 \right )} + \operatorname{atan}{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2 + 4*x + 5) + atan(x + 2)

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Mathematica [A]  time = 0.00830484, size = 14, normalized size = 1. \[ \log \left (x^2+4 x+5\right )+\tan ^{-1}(x+2) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[2 + x] + Log[5 + 4*x + x^2]

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Maple [A]  time = 0.005, size = 15, normalized size = 1.1 \[ \arctan \left ( 2+x \right ) +\ln \left ({x}^{2}+4\,x+5 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5+2*x)/(x^2+4*x+5),x)

[Out]

arctan(2+x)+ln(x^2+4*x+5)

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Maxima [A]  time = 0.761301, size = 19, normalized size = 1.36 \[ \arctan \left (x + 2\right ) + \log \left (x^{2} + 4 \, x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x + 2) + log(x^2 + 4*x + 5)

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Fricas [A]  time = 0.272558, size = 19, normalized size = 1.36 \[ \arctan \left (x + 2\right ) + \log \left (x^{2} + 4 \, x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x + 2) + log(x^2 + 4*x + 5)

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Sympy [A]  time = 0.194715, size = 14, normalized size = 1. \[ \log{\left (x^{2} + 4 x + 5 \right )} + \operatorname{atan}{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2 + 4*x + 5) + atan(x + 2)

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GIAC/XCAS [A]  time = 0.270474, size = 19, normalized size = 1.36 \[ \arctan \left (x + 2\right ) +{\rm ln}\left (x^{2} + 4 \, x + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x + 2) + ln(x^2 + 4*x + 5)