3.2408 \(\int \left (a+\frac{b}{\sqrt [3]{x}}\right )^3 x \, dx\)

Optimal. Leaf size=42 \[ \frac{a^3 x^2}{2}+\frac{9}{5} a^2 b x^{5/3}+\frac{9}{4} a b^2 x^{4/3}+b^3 x \]

[Out]

b^3*x + (9*a*b^2*x^(4/3))/4 + (9*a^2*b*x^(5/3))/5 + (a^3*x^2)/2

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Rubi [A]  time = 0.0670374, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a^3 x^2}{2}+\frac{9}{5} a^2 b x^{5/3}+\frac{9}{4} a b^2 x^{4/3}+b^3 x \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^3*x,x]

[Out]

b^3*x + (9*a*b^2*x^(4/3))/4 + (9*a^2*b*x^(5/3))/5 + (a^3*x^2)/2

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Rubi in Sympy [A]  time = 9.73014, size = 39, normalized size = 0.93 \[ \frac{a^{3} x^{2}}{2} + \frac{9 a^{2} b x^{\frac{5}{3}}}{5} + \frac{9 a b^{2} x^{\frac{4}{3}}}{4} + b^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**(1/3))**3*x,x)

[Out]

a**3*x**2/2 + 9*a**2*b*x**(5/3)/5 + 9*a*b**2*x**(4/3)/4 + b**3*x

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Mathematica [A]  time = 0.00955789, size = 42, normalized size = 1. \[ \frac{a^3 x^2}{2}+\frac{9}{5} a^2 b x^{5/3}+\frac{9}{4} a b^2 x^{4/3}+b^3 x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^(1/3))^3*x,x]

[Out]

b^3*x + (9*a*b^2*x^(4/3))/4 + (9*a^2*b*x^(5/3))/5 + (a^3*x^2)/2

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Maple [A]  time = 0.003, size = 33, normalized size = 0.8 \[{b}^{3}x+{\frac{9\,a{b}^{2}}{4}{x}^{{\frac{4}{3}}}}+{\frac{9\,{a}^{2}b}{5}{x}^{{\frac{5}{3}}}}+{\frac{{x}^{2}{a}^{3}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^3*x,x)

[Out]

b^3*x+9/4*a*b^2*x^(4/3)+9/5*a^2*b*x^(5/3)+1/2*x^2*a^3

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Maxima [A]  time = 1.43768, size = 50, normalized size = 1.19 \[ \frac{1}{20} \,{\left (10 \, a^{3} + \frac{36 \, a^{2} b}{x^{\frac{1}{3}}} + \frac{45 \, a b^{2}}{x^{\frac{2}{3}}} + \frac{20 \, b^{3}}{x}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^3*x,x, algorithm="maxima")

[Out]

1/20*(10*a^3 + 36*a^2*b/x^(1/3) + 45*a*b^2/x^(2/3) + 20*b^3/x)*x^2

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Fricas [A]  time = 0.217973, size = 43, normalized size = 1.02 \[ \frac{1}{2} \, a^{3} x^{2} + \frac{9}{5} \, a^{2} b x^{\frac{5}{3}} + \frac{9}{4} \, a b^{2} x^{\frac{4}{3}} + b^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^3*x,x, algorithm="fricas")

[Out]

1/2*a^3*x^2 + 9/5*a^2*b*x^(5/3) + 9/4*a*b^2*x^(4/3) + b^3*x

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Sympy [A]  time = 2.486, size = 39, normalized size = 0.93 \[ \frac{a^{3} x^{2}}{2} + \frac{9 a^{2} b x^{\frac{5}{3}}}{5} + \frac{9 a b^{2} x^{\frac{4}{3}}}{4} + b^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**(1/3))**3*x,x)

[Out]

a**3*x**2/2 + 9*a**2*b*x**(5/3)/5 + 9*a*b**2*x**(4/3)/4 + b**3*x

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GIAC/XCAS [A]  time = 0.211498, size = 43, normalized size = 1.02 \[ \frac{1}{2} \, a^{3} x^{2} + \frac{9}{5} \, a^{2} b x^{\frac{5}{3}} + \frac{9}{4} \, a b^{2} x^{\frac{4}{3}} + b^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^3*x,x, algorithm="giac")

[Out]

1/2*a^3*x^2 + 9/5*a^2*b*x^(5/3) + 9/4*a*b^2*x^(4/3) + b^3*x