3.1570 \(\int \left (a+\frac{b}{x}\right )^3 x^4 \, dx\)

Optimal. Leaf size=30 \[ \frac{(a x+b)^5}{5 a^2}-\frac{b (a x+b)^4}{4 a^2} \]

[Out]

-(b*(b + a*x)^4)/(4*a^2) + (b + a*x)^5/(5*a^2)

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Rubi [A]  time = 0.0391365, antiderivative size = 30, 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 \[ \frac{(a x+b)^5}{5 a^2}-\frac{b (a x+b)^4}{4 a^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^3*x^4,x]

[Out]

-(b*(b + a*x)^4)/(4*a^2) + (b + a*x)^5/(5*a^2)

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Rubi in Sympy [A]  time = 7.65092, size = 24, normalized size = 0.8 \[ - \frac{b \left (a x + b\right )^{4}}{4 a^{2}} + \frac{\left (a x + b\right )^{5}}{5 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**3*x**4,x)

[Out]

-b*(a*x + b)**4/(4*a**2) + (a*x + b)**5/(5*a**2)

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Mathematica [A]  time = 0.00262162, size = 40, normalized size = 1.33 \[ \frac{a^3 x^5}{5}+\frac{3}{4} a^2 b x^4+a b^2 x^3+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^3*x^4,x]

[Out]

(b^3*x^2)/2 + a*b^2*x^3 + (3*a^2*b*x^4)/4 + (a^3*x^5)/5

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Maple [A]  time = 0.001, size = 35, normalized size = 1.2 \[{\frac{{a}^{3}{x}^{5}}{5}}+{\frac{3\,{a}^{2}b{x}^{4}}{4}}+a{b}^{2}{x}^{3}+{\frac{{b}^{3}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^3*x^4,x)

[Out]

1/5*a^3*x^5+3/4*a^2*b*x^4+a*b^2*x^3+1/2*b^3*x^2

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Maxima [A]  time = 1.44378, size = 46, normalized size = 1.53 \[ \frac{1}{5} \, a^{3} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + a b^{2} x^{3} + \frac{1}{2} \, b^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3*x^4,x, algorithm="maxima")

[Out]

1/5*a^3*x^5 + 3/4*a^2*b*x^4 + a*b^2*x^3 + 1/2*b^3*x^2

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Fricas [A]  time = 0.209575, size = 46, normalized size = 1.53 \[ \frac{1}{5} \, a^{3} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + a b^{2} x^{3} + \frac{1}{2} \, b^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3*x^4,x, algorithm="fricas")

[Out]

1/5*a^3*x^5 + 3/4*a^2*b*x^4 + a*b^2*x^3 + 1/2*b^3*x^2

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Sympy [A]  time = 0.091865, size = 36, normalized size = 1.2 \[ \frac{a^{3} x^{5}}{5} + \frac{3 a^{2} b x^{4}}{4} + a b^{2} x^{3} + \frac{b^{3} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**3*x**4,x)

[Out]

a**3*x**5/5 + 3*a**2*b*x**4/4 + a*b**2*x**3 + b**3*x**2/2

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GIAC/XCAS [A]  time = 0.223652, size = 46, normalized size = 1.53 \[ \frac{1}{5} \, a^{3} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + a b^{2} x^{3} + \frac{1}{2} \, b^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^3*x^4,x, algorithm="giac")

[Out]

1/5*a^3*x^5 + 3/4*a^2*b*x^4 + a*b^2*x^3 + 1/2*b^3*x^2