3.443 \(\int \frac{x+2 x^3}{\left (x^2+x^4\right )^3} \, dx\)

Optimal. Leaf size=13 \[ -\frac{1}{4 \left (x^4+x^2\right )^2} \]

[Out]

-1/(4*(x^2 + x^4)^2)

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Rubi [A]  time = 0.0083106, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{1}{4 \left (x^4+x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*(x^2 + x^4)^2)

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Rubi in Sympy [A]  time = 8.47684, size = 14, normalized size = 1.08 \[ - \frac{1}{4 x^{4} \left (x^{2} + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(4*x**4*(x**2 + 1)**2)

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Mathematica [A]  time = 0.0102078, size = 14, normalized size = 1.08 \[ -\frac{1}{4 x^4 \left (x^2+1\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*x^4*(1 + x^2)^2)

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Maple [B]  time = 0.017, size = 30, normalized size = 2.3 \[ -{\frac{1}{4\,{x}^{4}}}+{\frac{1}{2\,{x}^{2}}}-{\frac{1}{4\, \left ({x}^{2}+1 \right ) ^{2}}}-{\frac{1}{2\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^3+x)/(x^4+x^2)^3,x)

[Out]

-1/4/x^4+1/2/x^2-1/4/(x^2+1)^2-1/2/(x^2+1)

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Maxima [A]  time = 0.788247, size = 15, normalized size = 1.15 \[ -\frac{1}{4 \,{\left (x^{4} + x^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/(x^4 + x^2)^2

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Fricas [A]  time = 0.244984, size = 22, normalized size = 1.69 \[ -\frac{1}{4 \,{\left (x^{8} + 2 \, x^{6} + x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/(x^8 + 2*x^6 + x^4)

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Sympy [A]  time = 0.367543, size = 17, normalized size = 1.31 \[ - \frac{1}{4 x^{8} + 8 x^{6} + 4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(4*x**8 + 8*x**6 + 4*x**4)

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GIAC/XCAS [A]  time = 0.259861, size = 15, normalized size = 1.15 \[ -\frac{1}{4 \,{\left (x^{4} + x^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/(x^4 + x^2)^2