3.1839 \(\int \frac{\left (a+\frac{b}{x^2}\right )^3}{x^5} \, dx\)

Optimal. Leaf size=40 \[ \frac{a \left (a x^2+b\right )^4}{40 b^2 x^8}-\frac{\left (a x^2+b\right )^4}{10 b x^{10}} \]

[Out]

-(b + a*x^2)^4/(10*b*x^10) + (a*(b + a*x^2)^4)/(40*b^2*x^8)

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Rubi [A]  time = 0.064114, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{a \left (a x^2+b\right )^4}{40 b^2 x^8}-\frac{\left (a x^2+b\right )^4}{10 b x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^3/x^5,x]

[Out]

-(b + a*x^2)^4/(10*b*x^10) + (a*(b + a*x^2)^4)/(40*b^2*x^8)

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Rubi in Sympy [A]  time = 10.1987, size = 39, normalized size = 0.98 \[ - \frac{a^{3}}{4 x^{4}} - \frac{a^{2} b}{2 x^{6}} - \frac{3 a b^{2}}{8 x^{8}} - \frac{b^{3}}{10 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**3/x**5,x)

[Out]

-a**3/(4*x**4) - a**2*b/(2*x**6) - 3*a*b**2/(8*x**8) - b**3/(10*x**10)

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Mathematica [A]  time = 0.00717978, size = 43, normalized size = 1.08 \[ -\frac{a^3}{4 x^4}-\frac{a^2 b}{2 x^6}-\frac{3 a b^2}{8 x^8}-\frac{b^3}{10 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^3/x^5,x]

[Out]

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

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Maple [A]  time = 0.007, size = 36, normalized size = 0.9 \[ -{\frac{{a}^{2}b}{2\,{x}^{6}}}-{\frac{{a}^{3}}{4\,{x}^{4}}}-{\frac{{b}^{3}}{10\,{x}^{10}}}-{\frac{3\,a{b}^{2}}{8\,{x}^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^3/x^5,x)

[Out]

-1/2*a^2*b/x^6-1/4*a^3/x^4-1/10*b^3/x^10-3/8*a*b^2/x^8

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Maxima [A]  time = 1.48167, size = 50, normalized size = 1.25 \[ -\frac{10 \, a^{3} x^{6} + 20 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2} + 4 \, b^{3}}{40 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3/x^5,x, algorithm="maxima")

[Out]

-1/40*(10*a^3*x^6 + 20*a^2*b*x^4 + 15*a*b^2*x^2 + 4*b^3)/x^10

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Fricas [A]  time = 0.219906, size = 50, normalized size = 1.25 \[ -\frac{10 \, a^{3} x^{6} + 20 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2} + 4 \, b^{3}}{40 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3/x^5,x, algorithm="fricas")

[Out]

-1/40*(10*a^3*x^6 + 20*a^2*b*x^4 + 15*a*b^2*x^2 + 4*b^3)/x^10

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Sympy [A]  time = 1.77216, size = 39, normalized size = 0.98 \[ - \frac{10 a^{3} x^{6} + 20 a^{2} b x^{4} + 15 a b^{2} x^{2} + 4 b^{3}}{40 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**3/x**5,x)

[Out]

-(10*a**3*x**6 + 20*a**2*b*x**4 + 15*a*b**2*x**2 + 4*b**3)/(40*x**10)

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GIAC/XCAS [A]  time = 0.221473, size = 50, normalized size = 1.25 \[ -\frac{10 \, a^{3} x^{6} + 20 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2} + 4 \, b^{3}}{40 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3/x^5,x, algorithm="giac")

[Out]

-1/40*(10*a^3*x^6 + 20*a^2*b*x^4 + 15*a*b^2*x^2 + 4*b^3)/x^10