3.480 \(\int \frac{x}{a^2+2 a b x^2+b^2 x^4} \, dx\)

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

[Out]

-1/(2*b*(a + b*x^2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0153704, 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 \[ -\frac{1}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*b*(a + b*x^2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.74245, size = 12, normalized size = 0.75 \[ - \frac{1}{2 b \left (a + b x^{2}\right )} \]

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),x)

[Out]

-1/(2*b*(a + b*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.00470919, size = 16, normalized size = 1. \[ -\frac{1}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*b*(a + b*x^2))

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 15, normalized size = 0.9 \[ -{\frac{1}{2\,b \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.696605, size = 20, normalized size = 1.25 \[ -\frac{1}{2 \,{\left (b^{2} x^{2} + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(b^2*x^2 + a*b)

_______________________________________________________________________________________

Fricas [A]  time = 0.246033, size = 20, normalized size = 1.25 \[ -\frac{1}{2 \,{\left (b^{2} x^{2} + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(b^2*x^2 + a*b)

_______________________________________________________________________________________

Sympy [A]  time = 1.15025, size = 15, normalized size = 0.94 \[ - \frac{1}{2 a b + 2 b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*a*b + 2*b**2*x**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270046, size = 19, normalized size = 1.19 \[ -\frac{1}{2 \,{\left (b x^{2} + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/((b*x^2 + a)*b)