3.1825 \(\int \frac{\left (a+\frac{b}{x^2}\right )^2}{x^4} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{3 x^3}-\frac{2 a b}{5 x^5}-\frac{b^2}{7 x^7} \]

[Out]

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

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Rubi [A]  time = 0.0427113, antiderivative size = 30, 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 \[ -\frac{a^2}{3 x^3}-\frac{2 a b}{5 x^5}-\frac{b^2}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^2/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.84685, size = 27, normalized size = 0.9 \[ - \frac{a^{2}}{3 x^{3}} - \frac{2 a b}{5 x^{5}} - \frac{b^{2}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**2/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.00149432, size = 30, normalized size = 1. \[ -\frac{a^2}{3 x^3}-\frac{2 a b}{5 x^5}-\frac{b^2}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^2/x^4,x]

[Out]

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

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Maple [A]  time = 0.007, size = 25, normalized size = 0.8 \[ -{\frac{{b}^{2}}{7\,{x}^{7}}}-{\frac{2\,ab}{5\,{x}^{5}}}-{\frac{{a}^{2}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^2/x^4,x)

[Out]

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

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Maxima [A]  time = 1.43876, size = 35, normalized size = 1.17 \[ -\frac{35 \, a^{2} x^{4} + 42 \, a b x^{2} + 15 \, b^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^2/x^4,x, algorithm="maxima")

[Out]

-1/105*(35*a^2*x^4 + 42*a*b*x^2 + 15*b^2)/x^7

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Fricas [A]  time = 0.219236, size = 35, normalized size = 1.17 \[ -\frac{35 \, a^{2} x^{4} + 42 \, a b x^{2} + 15 \, b^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^2/x^4,x, algorithm="fricas")

[Out]

-1/105*(35*a^2*x^4 + 42*a*b*x^2 + 15*b^2)/x^7

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Sympy [A]  time = 1.40507, size = 27, normalized size = 0.9 \[ - \frac{35 a^{2} x^{4} + 42 a b x^{2} + 15 b^{2}}{105 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**2/x**4,x)

[Out]

-(35*a**2*x**4 + 42*a*b*x**2 + 15*b**2)/(105*x**7)

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GIAC/XCAS [A]  time = 0.224848, size = 35, normalized size = 1.17 \[ -\frac{35 \, a^{2} x^{4} + 42 \, a b x^{2} + 15 \, b^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^2/x^4,x, algorithm="giac")

[Out]

-1/105*(35*a^2*x^4 + 42*a*b*x^2 + 15*b^2)/x^7