3.1573 \(\int \left (a+\frac{b}{x}\right )^3 x \, dx\)

Optimal. Leaf size=34 \[ \frac{a^3 x^2}{2}+3 a^2 b x+3 a b^2 \log (x)-\frac{b^3}{x} \]

[Out]

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

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Rubi [A]  time = 0.0464414, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{a^3 x^2}{2}+3 a^2 b x+3 a b^2 \log (x)-\frac{b^3}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^3*x,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**3*x,x)

[Out]

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

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Mathematica [A]  time = 0.00730713, size = 34, normalized size = 1. \[ \frac{a^3 x^2}{2}+3 a^2 b x+3 a b^2 \log (x)-\frac{b^3}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^3*x,x]

[Out]

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

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Maple [A]  time = 0.007, size = 33, normalized size = 1. \[ -{\frac{{b}^{3}}{x}}+3\,{a}^{2}bx+{\frac{{x}^{2}{a}^{3}}{2}}+3\,a{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^3*x,x)

[Out]

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

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Maxima [A]  time = 1.43974, size = 43, normalized size = 1.26 \[ \frac{1}{2} \, a^{3} x^{2} + 3 \, a^{2} b x + 3 \, a b^{2} \log \left (x\right ) - \frac{b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219111, size = 49, normalized size = 1.44 \[ \frac{a^{3} x^{3} + 6 \, a^{2} b x^{2} + 6 \, a b^{2} x \log \left (x\right ) - 2 \, b^{3}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(a^3*x^3 + 6*a^2*b*x^2 + 6*a*b^2*x*log(x) - 2*b^3)/x

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Sympy [A]  time = 1.13904, size = 31, normalized size = 0.91 \[ \frac{a^{3} x^{2}}{2} + 3 a^{2} b x + 3 a b^{2} \log{\left (x \right )} - \frac{b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**3*x,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.222165, size = 45, normalized size = 1.32 \[ \frac{1}{2} \, a^{3} x^{2} + 3 \, a^{2} b x + 3 \, a b^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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