3.905 \(\int \frac{4+x}{\left (5+4 x+x^2\right )^2} \, dx\)

Optimal. Leaf size=24 \[ \frac{2 x+3}{2 \left (x^2+4 x+5\right )}+\tan ^{-1}(x+2) \]

[Out]

(3 + 2*x)/(2*(5 + 4*x + x^2)) + ArcTan[2 + x]

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Rubi [A]  time = 0.0221041, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 x+3}{2 \left (x^2+4 x+5\right )}+\tan ^{-1}(x+2) \]

Antiderivative was successfully verified.

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

[Out]

(3 + 2*x)/(2*(5 + 4*x + x^2)) + ArcTan[2 + x]

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Rubi in Sympy [A]  time = 4.39601, size = 19, normalized size = 0.79 \[ \frac{4 x + 6}{4 \left (x^{2} + 4 x + 5\right )} + \operatorname{atan}{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(4*x + 6)/(4*(x**2 + 4*x + 5)) + atan(x + 2)

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Mathematica [A]  time = 0.0200706, size = 24, normalized size = 1. \[ \frac{2 x+3}{2 \left (x^2+4 x+5\right )}+\tan ^{-1}(x+2) \]

Antiderivative was successfully verified.

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

[Out]

(3 + 2*x)/(2*(5 + 4*x + x^2)) + ArcTan[2 + x]

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Maple [A]  time = 0.004, size = 23, normalized size = 1. \[{\frac{4\,x+6}{4\,{x}^{2}+16\,x+20}}+\arctan \left ( 2+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4+x)/(x^2+4*x+5)^2,x)

[Out]

1/4*(4*x+6)/(x^2+4*x+5)+arctan(2+x)

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Maxima [A]  time = 0.772106, size = 30, normalized size = 1.25 \[ \frac{2 \, x + 3}{2 \,{\left (x^{2} + 4 \, x + 5\right )}} + \arctan \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*x + 3)/(x^2 + 4*x + 5) + arctan(x + 2)

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Fricas [A]  time = 0.292004, size = 42, normalized size = 1.75 \[ \frac{2 \,{\left (x^{2} + 4 \, x + 5\right )} \arctan \left (x + 2\right ) + 2 \, x + 3}{2 \,{\left (x^{2} + 4 \, x + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*(x^2 + 4*x + 5)*arctan(x + 2) + 2*x + 3)/(x^2 + 4*x + 5)

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Sympy [A]  time = 0.260082, size = 19, normalized size = 0.79 \[ \frac{2 x + 3}{2 x^{2} + 8 x + 10} + \operatorname{atan}{\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*x + 3)/(2*x**2 + 8*x + 10) + atan(x + 2)

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GIAC/XCAS [A]  time = 0.268528, size = 30, normalized size = 1.25 \[ \frac{2 \, x + 3}{2 \,{\left (x^{2} + 4 \, x + 5\right )}} + \arctan \left (x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*x + 3)/(x^2 + 4*x + 5) + arctan(x + 2)