3.1638 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^3} \, dx\)

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

[Out]

-1/(2*a*(b + a*x)^2)

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Rubi [A]  time = 0.0168068, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{1}{2 a (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*x^3),x]

[Out]

-1/(2*a*(b + a*x)^2)

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Rubi in Sympy [A]  time = 2.90826, size = 12, normalized size = 0.86 \[ - \frac{1}{2 a \left (a x + b\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**3/x**3,x)

[Out]

-1/(2*a*(a*x + b)**2)

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Mathematica [A]  time = 0.00473479, size = 14, normalized size = 1. \[ -\frac{1}{2 a (a x+b)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x)^3*x^3),x]

[Out]

-1/(2*a*(b + a*x)^2)

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Maple [A]  time = 0.003, size = 13, normalized size = 0.9 \[ -{\frac{1}{2\,a \left ( ax+b \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x^3,x)

[Out]

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

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Maxima [A]  time = 1.44497, size = 32, normalized size = 2.29 \[ -\frac{1}{2 \,{\left (a^{3} x^{2} + 2 \, a^{2} b x + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(a^3*x^2 + 2*a^2*b*x + a*b^2)

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Fricas [A]  time = 0.214101, size = 32, normalized size = 2.29 \[ -\frac{1}{2 \,{\left (a^{3} x^{2} + 2 \, a^{2} b x + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(a^3*x^2 + 2*a^2*b*x + a*b^2)

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Sympy [A]  time = 1.30955, size = 26, normalized size = 1.86 \[ - \frac{1}{2 a^{3} x^{2} + 4 a^{2} b x + 2 a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**3/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.227487, size = 16, normalized size = 1.14 \[ -\frac{1}{2 \,{\left (a x + b\right )}^{2} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/((a*x + b)^2*a)