3.1558 \(\int \left (a+\frac{b}{x}\right )^2 x^4 \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^5}{5}+\frac{1}{2} a b x^4+\frac{b^2 x^3}{3} \]

[Out]

(b^2*x^3)/3 + (a*b*x^4)/2 + (a^2*x^5)/5

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Rubi [A]  time = 0.0423555, 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 \[ \frac{a^2 x^5}{5}+\frac{1}{2} a b x^4+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^2*x^4,x]

[Out]

(b^2*x^3)/3 + (a*b*x^4)/2 + (a^2*x^5)/5

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Rubi in Sympy [A]  time = 6.69783, size = 24, normalized size = 0.8 \[ \frac{a^{2} x^{5}}{5} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**5/5 + a*b*x**4/2 + b**2*x**3/3

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Mathematica [A]  time = 0.00285233, size = 30, normalized size = 1. \[ \frac{a^2 x^5}{5}+\frac{1}{2} a b x^4+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^2*x^4,x]

[Out]

(b^2*x^3)/3 + (a*b*x^4)/2 + (a^2*x^5)/5

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \[{\frac{{b}^{2}{x}^{3}}{3}}+{\frac{ab{x}^{4}}{2}}+{\frac{{x}^{5}{a}^{2}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^2*x^4,x)

[Out]

1/3*b^2*x^3+1/2*a*b*x^4+1/5*x^5*a^2

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Maxima [A]  time = 1.43979, size = 32, normalized size = 1.07 \[ \frac{1}{5} \, a^{2} x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, b^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*a^2*x^5 + 1/2*a*b*x^4 + 1/3*b^2*x^3

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Fricas [A]  time = 0.219653, size = 32, normalized size = 1.07 \[ \frac{1}{5} \, a^{2} x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, b^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*a^2*x^5 + 1/2*a*b*x^4 + 1/3*b^2*x^3

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Sympy [A]  time = 0.086104, size = 24, normalized size = 0.8 \[ \frac{a^{2} x^{5}}{5} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**2*x**4,x)

[Out]

a**2*x**5/5 + a*b*x**4/2 + b**2*x**3/3

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GIAC/XCAS [A]  time = 0.225928, size = 32, normalized size = 1.07 \[ \frac{1}{5} \, a^{2} x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, b^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*a^2*x^5 + 1/2*a*b*x^4 + 1/3*b^2*x^3