3.1817 \(\int \left (a+\frac{b}{x^2}\right )^2 x^4 \, dx\)

Optimal. Leaf size=25 \[ \frac{a^2 x^5}{5}+\frac{2}{3} a b x^3+b^2 x \]

[Out]

b^2*x + (2*a*b*x^3)/3 + (a^2*x^5)/5

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Rubi [A]  time = 0.0303961, antiderivative size = 25, 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 \[ \frac{a^2 x^5}{5}+\frac{2}{3} a b x^3+b^2 x \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^2*x^4,x]

[Out]

b^2*x + (2*a*b*x^3)/3 + (a^2*x^5)/5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} x^{5}}{5} + \frac{2 a b x^{3}}{3} + \int b^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**5/5 + 2*a*b*x**3/3 + Integral(b**2, x)

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Mathematica [A]  time = 0.00166711, size = 25, normalized size = 1. \[ \frac{a^2 x^5}{5}+\frac{2}{3} a b x^3+b^2 x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^2*x^4,x]

[Out]

b^2*x + (2*a*b*x^3)/3 + (a^2*x^5)/5

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Maple [A]  time = 0.001, size = 22, normalized size = 0.9 \[{b}^{2}x+{\frac{2\,ab{x}^{3}}{3}}+{\frac{{x}^{5}{a}^{2}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^2*x^4,x)

[Out]

b^2*x+2/3*a*b*x^3+1/5*x^5*a^2

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Maxima [A]  time = 1.44408, size = 28, normalized size = 1.12 \[ \frac{1}{5} \, a^{2} x^{5} + \frac{2}{3} \, a b x^{3} + b^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*a^2*x^5 + 2/3*a*b*x^3 + b^2*x

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Fricas [A]  time = 0.218809, size = 28, normalized size = 1.12 \[ \frac{1}{5} \, a^{2} x^{5} + \frac{2}{3} \, a b x^{3} + b^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*a^2*x^5 + 2/3*a*b*x^3 + b^2*x

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Sympy [A]  time = 0.085168, size = 22, normalized size = 0.88 \[ \frac{a^{2} x^{5}}{5} + \frac{2 a b x^{3}}{3} + b^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**2*x**4,x)

[Out]

a**2*x**5/5 + 2*a*b*x**3/3 + b**2*x

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GIAC/XCAS [A]  time = 0.227624, size = 28, normalized size = 1.12 \[ \frac{1}{5} \, a^{2} x^{5} + \frac{2}{3} \, a b x^{3} + b^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*a^2*x^5 + 2/3*a*b*x^3 + b^2*x