3.118 \(\int \frac{1+2 x^2}{x^5 \left (1+x^2\right )^3} \, dx\)

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

[Out]

-1/(4*x^4*(1 + x^2)^2)

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Rubi [A]  time = 0.0097358, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{1}{4 x^4 \left (x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(1 + 2*x^2)/(x^5*(1 + x^2)^3),x]

[Out]

-1/(4*x^4*(1 + x^2)^2)

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Rubi in Sympy [A]  time = 6.69551, size = 14, normalized size = 1. \[ - \frac{1}{4 x^{4} \left (x^{2} + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**2+1)/x**5/(x**2+1)**3,x)

[Out]

-1/(4*x**4*(x**2 + 1)**2)

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Mathematica [A]  time = 0.00977804, size = 14, normalized size = 1. \[ -\frac{1}{4 x^4 \left (x^2+1\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x^2)/(x^5*(1 + x^2)^3),x]

[Out]

-1/(4*x^4*(1 + x^2)^2)

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Maple [B]  time = 0.016, size = 30, normalized size = 2.1 \[ -{\frac{1}{4\,{x}^{4}}}+{\frac{1}{2\,{x}^{2}}}-{\frac{1}{4\, \left ({x}^{2}+1 \right ) ^{2}}}-{\frac{1}{2\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^2+1)/x^5/(x^2+1)^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 = 1.34297, size = 22, normalized size = 1.57 \[ -\frac{1}{4 \,{\left (x^{8} + 2 \, x^{6} + x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 1)/((x^2 + 1)^3*x^5),x, algorithm="maxima")

[Out]

-1/4/(x^8 + 2*x^6 + x^4)

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Fricas [A]  time = 0.203568, size = 22, normalized size = 1.57 \[ -\frac{1}{4 \,{\left (x^{8} + 2 \, x^{6} + x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 1)/((x^2 + 1)^3*x^5),x, algorithm="fricas")

[Out]

-1/4/(x^8 + 2*x^6 + x^4)

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Sympy [A]  time = 0.357549, size = 17, normalized size = 1.21 \[ - \frac{1}{4 x^{8} + 8 x^{6} + 4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**2+1)/x**5/(x**2+1)**3,x)

[Out]

-1/(4*x**8 + 8*x**6 + 4*x**4)

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GIAC/XCAS [A]  time = 0.227022, size = 15, normalized size = 1.07 \[ -\frac{1}{4 \,{\left (x^{4} + x^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^2 + 1)/((x^2 + 1)^3*x^5),x, algorithm="giac")

[Out]

-1/4/(x^4 + x^2)^2