3.2405 \(\int \left (a+\frac{b}{\sqrt [3]{x}}\right )^3 x^4 \, dx\)

Optimal. Leaf size=47 \[ \frac{a^3 x^5}{5}+\frac{9}{14} a^2 b x^{14/3}+\frac{9}{13} a b^2 x^{13/3}+\frac{b^3 x^4}{4} \]

[Out]

(b^3*x^4)/4 + (9*a*b^2*x^(13/3))/13 + (9*a^2*b*x^(14/3))/14 + (a^3*x^5)/5

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Rubi [A]  time = 0.0933029, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{a^3 x^5}{5}+\frac{9}{14} a^2 b x^{14/3}+\frac{9}{13} a b^2 x^{13/3}+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^3*x^4,x]

[Out]

(b^3*x^4)/4 + (9*a*b^2*x^(13/3))/13 + (9*a^2*b*x^(14/3))/14 + (a^3*x^5)/5

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Rubi in Sympy [A]  time = 14.6472, size = 42, normalized size = 0.89 \[ \frac{a^{3} x^{5}}{5} + \frac{9 a^{2} b x^{\frac{14}{3}}}{14} + \frac{9 a b^{2} x^{\frac{13}{3}}}{13} + \frac{b^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**(1/3))**3*x**4,x)

[Out]

a**3*x**5/5 + 9*a**2*b*x**(14/3)/14 + 9*a*b**2*x**(13/3)/13 + b**3*x**4/4

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Mathematica [A]  time = 0.0142908, size = 47, normalized size = 1. \[ \frac{a^3 x^5}{5}+\frac{9}{14} a^2 b x^{14/3}+\frac{9}{13} a b^2 x^{13/3}+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^(1/3))^3*x^4,x]

[Out]

(b^3*x^4)/4 + (9*a*b^2*x^(13/3))/13 + (9*a^2*b*x^(14/3))/14 + (a^3*x^5)/5

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Maple [A]  time = 0.002, size = 36, normalized size = 0.8 \[{\frac{{b}^{3}{x}^{4}}{4}}+{\frac{9\,a{b}^{2}}{13}{x}^{{\frac{13}{3}}}}+{\frac{9\,{a}^{2}b}{14}{x}^{{\frac{14}{3}}}}+{\frac{{a}^{3}{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^3*x^4,x)

[Out]

1/4*b^3*x^4+9/13*a*b^2*x^(13/3)+9/14*a^2*b*x^(14/3)+1/5*a^3*x^5

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Maxima [A]  time = 1.43955, size = 50, normalized size = 1.06 \[ \frac{1}{1820} \,{\left (364 \, a^{3} + \frac{1170 \, a^{2} b}{x^{\frac{1}{3}}} + \frac{1260 \, a b^{2}}{x^{\frac{2}{3}}} + \frac{455 \, b^{3}}{x}\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^3*x^4,x, algorithm="maxima")

[Out]

1/1820*(364*a^3 + 1170*a^2*b/x^(1/3) + 1260*a*b^2/x^(2/3) + 455*b^3/x)*x^5

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Fricas [A]  time = 0.21962, size = 47, normalized size = 1. \[ \frac{1}{5} \, a^{3} x^{5} + \frac{9}{14} \, a^{2} b x^{\frac{14}{3}} + \frac{9}{13} \, a b^{2} x^{\frac{13}{3}} + \frac{1}{4} \, b^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^3*x^4,x, algorithm="fricas")

[Out]

1/5*a^3*x^5 + 9/14*a^2*b*x^(14/3) + 9/13*a*b^2*x^(13/3) + 1/4*b^3*x^4

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Sympy [A]  time = 14.4227, size = 42, normalized size = 0.89 \[ \frac{a^{3} x^{5}}{5} + \frac{9 a^{2} b x^{\frac{14}{3}}}{14} + \frac{9 a b^{2} x^{\frac{13}{3}}}{13} + \frac{b^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**(1/3))**3*x**4,x)

[Out]

a**3*x**5/5 + 9*a**2*b*x**(14/3)/14 + 9*a*b**2*x**(13/3)/13 + b**3*x**4/4

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GIAC/XCAS [A]  time = 0.210331, size = 47, normalized size = 1. \[ \frac{1}{5} \, a^{3} x^{5} + \frac{9}{14} \, a^{2} b x^{\frac{14}{3}} + \frac{9}{13} \, a b^{2} x^{\frac{13}{3}} + \frac{1}{4} \, b^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^3*x^4,x, algorithm="giac")

[Out]

1/5*a^3*x^5 + 9/14*a^2*b*x^(14/3) + 9/13*a*b^2*x^(13/3) + 1/4*b^3*x^4