3.1637 \(\int \frac{1}{\left (a+\frac{b}{x}\right )^3 x^2} \, dx\)

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

[Out]

1/(2*b*(a + b/x)^2)

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Rubi [A]  time = 0.0166769, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{1}{2 b \left (a+\frac{b}{x}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x)^3*x^2),x]

[Out]

1/(2*b*(a + b/x)^2)

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Rubi in Sympy [A]  time = 2.20282, size = 10, normalized size = 0.62 \[ \frac{1}{2 b \left (a + \frac{b}{x}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*b*(a + b/x)**2)

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Mathematica [A]  time = 0.00955149, size = 20, normalized size = 1.25 \[ -\frac{2 a x+b}{2 a^2 (a x+b)^2} \]

Antiderivative was successfully verified.

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

[Out]

-(b + 2*a*x)/(2*a^2*(b + a*x)^2)

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Maple [A]  time = 0.007, size = 27, normalized size = 1.7 \[ -{\frac{1}{ \left ( ax+b \right ){a}^{2}}}+{\frac{b}{2\,{a}^{2} \left ( ax+b \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x)^3/x^2,x)

[Out]

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

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Maxima [A]  time = 1.41585, size = 19, normalized size = 1.19 \[ \frac{1}{2 \,{\left (a + \frac{b}{x}\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/((a + b/x)^2*b)

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Fricas [A]  time = 0.215087, size = 43, normalized size = 2.69 \[ -\frac{2 \, a x + b}{2 \,{\left (a^{4} x^{2} + 2 \, a^{3} b x + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.36549, size = 32, normalized size = 2. \[ - \frac{2 a x + b}{2 a^{4} x^{2} + 4 a^{3} b x + 2 a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x)**3/x**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.226367, size = 19, normalized size = 1.19 \[ \frac{1}{2 \,{\left (a + \frac{b}{x}\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/((a + b/x)^2*b)