3.1572 \(\int \left (a+\frac{b}{x}\right )^3 x^2 \, dx\)

Optimal. Leaf size=35 \[ \frac{a^3 x^3}{3}+\frac{3}{2} a^2 b x^2+3 a b^2 x+b^3 \log (x) \]

[Out]

3*a*b^2*x + (3*a^2*b*x^2)/2 + (a^3*x^3)/3 + b^3*Log[x]

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Rubi [A]  time = 0.0501241, antiderivative size = 35, 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 \[ \frac{a^3 x^3}{3}+\frac{3}{2} a^2 b x^2+3 a b^2 x+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^3*x^2,x]

[Out]

3*a*b^2*x + (3*a^2*b*x^2)/2 + (a^3*x^3)/3 + b^3*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00482502, size = 35, normalized size = 1. \[ \frac{a^3 x^3}{3}+\frac{3}{2} a^2 b x^2+3 a b^2 x+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^3*x^2,x]

[Out]

3*a*b^2*x + (3*a^2*b*x^2)/2 + (a^3*x^3)/3 + b^3*Log[x]

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Maple [A]  time = 0.003, size = 32, normalized size = 0.9 \[ 3\,a{b}^{2}x+{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{\frac{{a}^{3}{x}^{3}}{3}}+{b}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^3*x^2,x)

[Out]

3*a*b^2*x+3/2*a^2*b*x^2+1/3*a^3*x^3+b^3*ln(x)

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Maxima [A]  time = 1.44423, size = 42, normalized size = 1.2 \[ \frac{1}{3} \, a^{3} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^3*x^3 + 3/2*a^2*b*x^2 + 3*a*b^2*x + b^3*log(x)

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Fricas [A]  time = 0.221498, size = 42, normalized size = 1.2 \[ \frac{1}{3} \, a^{3} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^3*x^3 + 3/2*a^2*b*x^2 + 3*a*b^2*x + b^3*log(x)

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Sympy [A]  time = 1.07052, size = 34, normalized size = 0.97 \[ \frac{a^{3} x^{3}}{3} + \frac{3 a^{2} b x^{2}}{2} + 3 a b^{2} x + b^{3} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**3*x**2,x)

[Out]

a**3*x**3/3 + 3*a**2*b*x**2/2 + 3*a*b**2*x + b**3*log(x)

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GIAC/XCAS [A]  time = 0.226247, size = 43, normalized size = 1.23 \[ \frac{1}{3} \, a^{3} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^3*x^3 + 3/2*a^2*b*x^2 + 3*a*b^2*x + b^3*ln(abs(x))