3.1559 \(\int \left (a+\frac{b}{x}\right )^2 x^3 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0396504, 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 \[ \frac{a^2 x^4}{4}+\frac{2}{3} a b x^3+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^2*x^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} x^{4}}{4} + \frac{2 a b x^{3}}{3} + b^{2} \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**4/4 + 2*a*b*x**3/3 + b**2*Integral(x, x)

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Mathematica [A]  time = 0.0021698, size = 30, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{2}{3} a b x^3+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^2*x^3,x]

[Out]

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

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Maple [A]  time = 0.002, size = 25, normalized size = 0.8 \[{\frac{{b}^{2}{x}^{2}}{2}}+{\frac{2\,ab{x}^{3}}{3}}+{\frac{{x}^{4}{a}^{2}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^2*x^3,x)

[Out]

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

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Maxima [A]  time = 1.42069, size = 32, normalized size = 1.07 \[ \frac{1}{4} \, a^{2} x^{4} + \frac{2}{3} \, a b x^{3} + \frac{1}{2} \, b^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218063, size = 32, normalized size = 1.07 \[ \frac{1}{4} \, a^{2} x^{4} + \frac{2}{3} \, a b x^{3} + \frac{1}{2} \, b^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.094089, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{4}}{4} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**2*x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.22252, size = 32, normalized size = 1.07 \[ \frac{1}{4} \, a^{2} x^{4} + \frac{2}{3} \, a b x^{3} + \frac{1}{2} \, b^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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