3.440 \(\int \frac{2+4 x}{x^2+2 x^3+x^4} \, dx\)

Optimal. Leaf size=10 \[ -\frac{2}{x (x+1)} \]

[Out]

-2/(x*(1 + x))

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Rubi [A]  time = 0.0158958, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{2}{x (x+1)} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 4*x)/(x^2 + 2*x^3 + x^4),x]

[Out]

-2/(x*(1 + x))

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Rubi in Sympy [A]  time = 6.90936, size = 7, normalized size = 0.7 \[ - \frac{2}{x \left (x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+4*x)/(x**4+2*x**3+x**2),x)

[Out]

-2/(x*(x + 1))

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Mathematica [A]  time = 0.00890193, size = 9, normalized size = 0.9 \[ -\frac{2}{x^2+x} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + 4*x)/(x^2 + 2*x^3 + x^4),x]

[Out]

-2/(x + x^2)

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Maple [A]  time = 0.007, size = 14, normalized size = 1.4 \[ 2\, \left ( 1+x \right ) ^{-1}-2\,{x}^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+4*x)/(x^4+2*x^3+x^2),x)

[Out]

2/(1+x)-2/x

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Maxima [A]  time = 0.791325, size = 12, normalized size = 1.2 \[ -\frac{2}{x^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(2*x + 1)/(x^4 + 2*x^3 + x^2),x, algorithm="maxima")

[Out]

-2/(x^2 + x)

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Fricas [A]  time = 0.25147, size = 12, normalized size = 1.2 \[ -\frac{2}{x^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(2*x + 1)/(x^4 + 2*x^3 + x^2),x, algorithm="fricas")

[Out]

-2/(x^2 + x)

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Sympy [A]  time = 0.161714, size = 7, normalized size = 0.7 \[ - \frac{2}{x^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+4*x)/(x**4+2*x**3+x**2),x)

[Out]

-2/(x**2 + x)

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GIAC/XCAS [A]  time = 0.258634, size = 12, normalized size = 1.2 \[ -\frac{2}{x^{2} + x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2*(2*x + 1)/(x^4 + 2*x^3 + x^2),x, algorithm="giac")

[Out]

-2/(x^2 + x)