3.2396 \(\int \left (a+\frac{b}{\sqrt [3]{x}}\right )^2 x^4 \, dx\)

Optimal. Leaf size=34 \[ \frac{a^2 x^5}{5}+\frac{3}{7} a b x^{14/3}+\frac{3}{13} b^2 x^{13/3} \]

[Out]

(3*b^2*x^(13/3))/13 + (3*a*b*x^(14/3))/7 + (a^2*x^5)/5

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Rubi [A]  time = 0.0811189, antiderivative size = 34, 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 \[ \frac{a^2 x^5}{5}+\frac{3}{7} a b x^{14/3}+\frac{3}{13} b^2 x^{13/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^2*x^4,x]

[Out]

(3*b^2*x^(13/3))/13 + (3*a*b*x^(14/3))/7 + (a^2*x^5)/5

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Rubi in Sympy [A]  time = 12.8685, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{5}}{5} + \frac{3 a b x^{\frac{14}{3}}}{7} + \frac{3 b^{2} x^{\frac{13}{3}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**(1/3))**2*x**4,x)

[Out]

a**2*x**5/5 + 3*a*b*x**(14/3)/7 + 3*b**2*x**(13/3)/13

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Mathematica [A]  time = 0.014837, size = 34, normalized size = 1. \[ \frac{a^2 x^5}{5}+\frac{3}{7} a b x^{14/3}+\frac{3}{13} b^2 x^{13/3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^(1/3))^2*x^4,x]

[Out]

(3*b^2*x^(13/3))/13 + (3*a*b*x^(14/3))/7 + (a^2*x^5)/5

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Maple [A]  time = 0.001, size = 25, normalized size = 0.7 \[{\frac{3\,{b}^{2}}{13}{x}^{{\frac{13}{3}}}}+{\frac{3\,ab}{7}{x}^{{\frac{14}{3}}}}+{\frac{{x}^{5}{a}^{2}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^2*x^4,x)

[Out]

3/13*b^2*x^(13/3)+3/7*a*b*x^(14/3)+1/5*x^5*a^2

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Maxima [A]  time = 1.44566, size = 35, normalized size = 1.03 \[ \frac{1}{455} \,{\left (91 \, a^{2} + \frac{195 \, a b}{x^{\frac{1}{3}}} + \frac{105 \, b^{2}}{x^{\frac{2}{3}}}\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^2*x^4,x, algorithm="maxima")

[Out]

1/455*(91*a^2 + 195*a*b/x^(1/3) + 105*b^2/x^(2/3))*x^5

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Fricas [A]  time = 0.217415, size = 32, normalized size = 0.94 \[ \frac{1}{5} \, a^{2} x^{5} + \frac{3}{7} \, a b x^{\frac{14}{3}} + \frac{3}{13} \, b^{2} x^{\frac{13}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^2*x^4,x, algorithm="fricas")

[Out]

1/5*a^2*x^5 + 3/7*a*b*x^(14/3) + 3/13*b^2*x^(13/3)

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Sympy [A]  time = 11.6249, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{5}}{5} + \frac{3 a b x^{\frac{14}{3}}}{7} + \frac{3 b^{2} x^{\frac{13}{3}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**(1/3))**2*x**4,x)

[Out]

a**2*x**5/5 + 3*a*b*x**(14/3)/7 + 3*b**2*x**(13/3)/13

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GIAC/XCAS [A]  time = 0.209569, size = 32, normalized size = 0.94 \[ \frac{1}{5} \, a^{2} x^{5} + \frac{3}{7} \, a b x^{\frac{14}{3}} + \frac{3}{13} \, b^{2} x^{\frac{13}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^2*x^4,x, algorithm="giac")

[Out]

1/5*a^2*x^5 + 3/7*a*b*x^(14/3) + 3/13*b^2*x^(13/3)