3.1836 \(\int \frac{\left (a+\frac{b}{x^2}\right )^3}{x^2} \, dx\)

Optimal. Leaf size=39 \[ -\frac{a^3}{x}-\frac{a^2 b}{x^3}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{7 x^7} \]

[Out]

-b^3/(7*x^7) - (3*a*b^2)/(5*x^5) - (a^2*b)/x^3 - a^3/x

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Rubi [A]  time = 0.0505893, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^3}{x}-\frac{a^2 b}{x^3}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^3/x^2,x]

[Out]

-b^3/(7*x^7) - (3*a*b^2)/(5*x^5) - (a^2*b)/x^3 - a^3/x

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Rubi in Sympy [A]  time = 8.63712, size = 34, normalized size = 0.87 \[ - \frac{a^{3}}{x} - \frac{a^{2} b}{x^{3}} - \frac{3 a b^{2}}{5 x^{5}} - \frac{b^{3}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3/x - a**2*b/x**3 - 3*a*b**2/(5*x**5) - b**3/(7*x**7)

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Mathematica [A]  time = 0.00700091, size = 39, normalized size = 1. \[ -\frac{a^3}{x}-\frac{a^2 b}{x^3}-\frac{3 a b^2}{5 x^5}-\frac{b^3}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^3/x^2,x]

[Out]

-b^3/(7*x^7) - (3*a*b^2)/(5*x^5) - (a^2*b)/x^3 - a^3/x

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Maple [A]  time = 0.007, size = 36, normalized size = 0.9 \[ -{\frac{{b}^{3}}{7\,{x}^{7}}}-{\frac{3\,a{b}^{2}}{5\,{x}^{5}}}-{\frac{{a}^{2}b}{{x}^{3}}}-{\frac{{a}^{3}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^3/x^2,x)

[Out]

-1/7*b^3/x^7-3/5*a*b^2/x^5-a^2*b/x^3-a^3/x

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Maxima [A]  time = 1.43254, size = 50, normalized size = 1.28 \[ -\frac{35 \, a^{3} x^{6} + 35 \, a^{2} b x^{4} + 21 \, a b^{2} x^{2} + 5 \, b^{3}}{35 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(35*a^3*x^6 + 35*a^2*b*x^4 + 21*a*b^2*x^2 + 5*b^3)/x^7

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Fricas [A]  time = 0.210301, size = 50, normalized size = 1.28 \[ -\frac{35 \, a^{3} x^{6} + 35 \, a^{2} b x^{4} + 21 \, a b^{2} x^{2} + 5 \, b^{3}}{35 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(35*a^3*x^6 + 35*a^2*b*x^4 + 21*a*b^2*x^2 + 5*b^3)/x^7

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Sympy [A]  time = 1.5551, size = 39, normalized size = 1. \[ - \frac{35 a^{3} x^{6} + 35 a^{2} b x^{4} + 21 a b^{2} x^{2} + 5 b^{3}}{35 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**3/x**2,x)

[Out]

-(35*a**3*x**6 + 35*a**2*b*x**4 + 21*a*b**2*x**2 + 5*b**3)/(35*x**7)

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GIAC/XCAS [A]  time = 0.22975, size = 50, normalized size = 1.28 \[ -\frac{35 \, a^{3} x^{6} + 35 \, a^{2} b x^{4} + 21 \, a b^{2} x^{2} + 5 \, b^{3}}{35 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/35*(35*a^3*x^6 + 35*a^2*b*x^4 + 21*a*b^2*x^2 + 5*b^3)/x^7