3.1576 \(\int \frac{\left (a+\frac{b}{x}\right )^3}{x^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\left (a+\frac{b}{x}\right )^4}{4 b} \]

[Out]

-(a + b/x)^4/(4*b)

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Rubi [A]  time = 0.0160148, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\left (a+\frac{b}{x}\right )^4}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^3/x^2,x]

[Out]

-(a + b/x)^4/(4*b)

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Rubi in Sympy [A]  time = 2.19363, size = 10, normalized size = 0.62 \[ - \frac{\left (a + \frac{b}{x}\right )^{4}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b/x)**4/(4*b)

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Mathematica [B]  time = 0.00605344, size = 39, normalized size = 2.44 \[ -\frac{a^3}{x}-\frac{3 a^2 b}{2 x^2}-\frac{a b^2}{x^3}-\frac{b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^3/x^2,x]

[Out]

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

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Maple [B]  time = 0.007, size = 36, normalized size = 2.3 \[ -{\frac{{b}^{3}}{4\,{x}^{4}}}-{\frac{a{b}^{2}}{{x}^{3}}}-{\frac{3\,{a}^{2}b}{2\,{x}^{2}}}-{\frac{{a}^{3}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^3/x^2,x)

[Out]

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

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Maxima [A]  time = 1.44028, size = 19, normalized size = 1.19 \[ -\frac{{\left (a + \frac{b}{x}\right )}^{4}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(a + b/x)^4/b

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Fricas [A]  time = 0.209471, size = 45, normalized size = 2.81 \[ -\frac{4 \, a^{3} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a b^{2} x + b^{3}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(4*a^3*x^3 + 6*a^2*b*x^2 + 4*a*b^2*x + b^3)/x^4

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Sympy [A]  time = 1.4168, size = 36, normalized size = 2.25 \[ - \frac{4 a^{3} x^{3} + 6 a^{2} b x^{2} + 4 a b^{2} x + b^{3}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**3/x**2,x)

[Out]

-(4*a**3*x**3 + 6*a**2*b*x**2 + 4*a*b**2*x + b**3)/(4*x**4)

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GIAC/XCAS [A]  time = 0.229494, size = 19, normalized size = 1.19 \[ -\frac{{\left (a + \frac{b}{x}\right )}^{4}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(a + b/x)^4/b