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

Optimal. Leaf size=16 \[ -\frac{1}{4 b \left (a+b x^2\right )^2} \]

[Out]

-1/(4*b*(a + b*x^2)^2)

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Rubi [A]  time = 0.0109076, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{1}{4 b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^2)^3,x]

[Out]

-1/(4*b*(a + b*x^2)^2)

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Rubi in Sympy [A]  time = 2.2242, size = 14, normalized size = 0.88 \[ - \frac{1}{4 b \left (a + b x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(4*b*(a + b*x**2)**2)

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Mathematica [A]  time = 0.00447464, size = 16, normalized size = 1. \[ -\frac{1}{4 b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x^2)^3,x]

[Out]

-1/(4*b*(a + b*x^2)^2)

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Maple [A]  time = 0.002, size = 15, normalized size = 0.9 \[ -{\frac{1}{4\,b \left ( b{x}^{2}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^2+a)^3,x)

[Out]

-1/4/b/(b*x^2+a)^2

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Maxima [A]  time = 1.34332, size = 19, normalized size = 1.19 \[ -\frac{1}{4 \,{\left (b x^{2} + a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/((b*x^2 + a)^2*b)

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Fricas [A]  time = 0.222624, size = 35, normalized size = 2.19 \[ -\frac{1}{4 \,{\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/(b^3*x^4 + 2*a*b^2*x^2 + a^2*b)

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Sympy [A]  time = 1.52871, size = 27, normalized size = 1.69 \[ - \frac{1}{4 a^{2} b + 8 a b^{2} x^{2} + 4 b^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(b*x**2+a)**3,x)

[Out]

-1/(4*a**2*b + 8*a*b**2*x**2 + 4*b**3*x**4)

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GIAC/XCAS [A]  time = 0.210801, size = 19, normalized size = 1.19 \[ -\frac{1}{4 \,{\left (b x^{2} + a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4/((b*x^2 + a)^2*b)