3.1548 \(\int \left (a+\frac{b}{x}\right ) x^2 \, dx\)

Optimal. Leaf size=17 \[ \frac{a x^3}{3}+\frac{b x^2}{2} \]

[Out]

(b*x^2)/2 + (a*x^3)/3

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Rubi [A]  time = 0.0182768, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a x^3}{3}+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)*x^2,x]

[Out]

(b*x^2)/2 + (a*x^3)/3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)*x**2,x)

[Out]

a*x**3/3 + b*Integral(x, x)

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Mathematica [A]  time = 0.00161591, size = 17, normalized size = 1. \[ \frac{a x^3}{3}+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)*x^2,x]

[Out]

(b*x^2)/2 + (a*x^3)/3

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Maple [A]  time = 0.001, size = 14, normalized size = 0.8 \[{\frac{b{x}^{2}}{2}}+{\frac{a{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)*x^2,x)

[Out]

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

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Maxima [A]  time = 1.42506, size = 18, normalized size = 1.06 \[ \frac{1}{3} \, a x^{3} + \frac{1}{2} \, b x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a*x^3 + 1/2*b*x^2

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Fricas [A]  time = 0.21458, size = 18, normalized size = 1.06 \[ \frac{1}{3} \, a x^{3} + \frac{1}{2} \, b x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a*x^3 + 1/2*b*x^2

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Sympy [A]  time = 0.065123, size = 12, normalized size = 0.71 \[ \frac{a x^{3}}{3} + \frac{b x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)*x**2,x)

[Out]

a*x**3/3 + b*x**2/2

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GIAC/XCAS [A]  time = 0.22597, size = 18, normalized size = 1.06 \[ \frac{1}{3} \, a x^{3} + \frac{1}{2} \, b x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a*x^3 + 1/2*b*x^2