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

Optimal. Leaf size=16 \[ -\frac{1}{10 b \left (a+b x^2\right )^5} \]

[Out]

-1/(10*b*(a + b*x^2)^5)

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Rubi [A]  time = 0.0142325, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{1}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

-1/(10*b*(a + b*x^2)^5)

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Rubi in Sympy [A]  time = 6.31496, size = 14, normalized size = 0.88 \[ - \frac{1}{10 b \left (a + b x^{2}\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(10*b*(a + b*x**2)**5)

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Mathematica [A]  time = 0.00526628, size = 16, normalized size = 1. \[ -\frac{1}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

-1/(10*b*(a + b*x^2)^5)

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Maple [A]  time = 0.006, size = 15, normalized size = 0.9 \[ -{\frac{1}{10\,b \left ( b{x}^{2}+a \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-1/10/b/(b*x^2+a)^5

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Maxima [A]  time = 0.689691, size = 80, normalized size = 5. \[ -\frac{1}{10 \,{\left (b^{6} x^{10} + 5 \, a b^{5} x^{8} + 10 \, a^{2} b^{4} x^{6} + 10 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{2} + a^{5} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10/(b^6*x^10 + 5*a*b^5*x^8 + 10*a^2*b^4*x^6 + 10*a^3*b^3*x^4 + 5*a^4*b^2*x^2
+ a^5*b)

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Fricas [A]  time = 0.248255, size = 80, normalized size = 5. \[ -\frac{1}{10 \,{\left (b^{6} x^{10} + 5 \, a b^{5} x^{8} + 10 \, a^{2} b^{4} x^{6} + 10 \, a^{3} b^{3} x^{4} + 5 \, a^{4} b^{2} x^{2} + a^{5} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10/(b^6*x^10 + 5*a*b^5*x^8 + 10*a^2*b^4*x^6 + 10*a^3*b^3*x^4 + 5*a^4*b^2*x^2
+ a^5*b)

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Sympy [A]  time = 3.64774, size = 63, normalized size = 3.94 \[ - \frac{1}{10 a^{5} b + 50 a^{4} b^{2} x^{2} + 100 a^{3} b^{3} x^{4} + 100 a^{2} b^{4} x^{6} + 50 a b^{5} x^{8} + 10 b^{6} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(10*a**5*b + 50*a**4*b**2*x**2 + 100*a**3*b**3*x**4 + 100*a**2*b**4*x**6 + 50
*a*b**5*x**8 + 10*b**6*x**10)

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GIAC/XCAS [A]  time = 0.269741, size = 19, normalized size = 1.19 \[ -\frac{1}{10 \,{\left (b x^{2} + a\right )}^{5} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10/((b*x^2 + a)^5*b)