3.1560 \(\int \left (a+\frac{b}{x}\right )^2 x^2 \, dx\)

Optimal. Leaf size=14 \[ \frac{(a x+b)^3}{3 a} \]

[Out]

(b + a*x)^3/(3*a)

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Rubi [A]  time = 0.0164609, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(a x+b)^3}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^2*x^2,x]

[Out]

(b + a*x)^3/(3*a)

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Rubi in Sympy [A]  time = 2.96425, size = 8, normalized size = 0.57 \[ \frac{\left (a x + b\right )^{3}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*x + b)**3/(3*a)

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Mathematica [A]  time = 0.00283409, size = 14, normalized size = 1. \[ \frac{(a x+b)^3}{3 a} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^2*x^2,x]

[Out]

(b + a*x)^3/(3*a)

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Maple [A]  time = 0.001, size = 13, normalized size = 0.9 \[{\frac{ \left ( ax+b \right ) ^{3}}{3\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^2*x^2,x)

[Out]

1/3*(a*x+b)^3/a

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Maxima [A]  time = 1.446, size = 27, normalized size = 1.93 \[ \frac{1}{3} \, a^{2} x^{3} + a b x^{2} + b^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^2*x^3 + a*b*x^2 + b^2*x

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Fricas [A]  time = 0.217358, size = 27, normalized size = 1.93 \[ \frac{1}{3} \, a^{2} x^{3} + a b x^{2} + b^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^2*x^3 + a*b*x^2 + b^2*x

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Sympy [A]  time = 0.085349, size = 19, normalized size = 1.36 \[ \frac{a^{2} x^{3}}{3} + a b x^{2} + b^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**2*x**2,x)

[Out]

a**2*x**3/3 + a*b*x**2 + b**2*x

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GIAC/XCAS [A]  time = 0.222092, size = 27, normalized size = 1.93 \[ \frac{1}{3} \, a^{2} x^{3} + a b x^{2} + b^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^2*x^3 + a*b*x^2 + b^2*x