3.2390 \(\int \left (a+\frac{b}{\sqrt [3]{x}}\right ) x \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^2}{2}+\frac{3}{5} b x^{5/3} \]

[Out]

(3*b*x^(5/3))/5 + (a*x^2)/2

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Rubi [A]  time = 0.0157675, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a x^2}{2}+\frac{3}{5} b x^{5/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))*x,x]

[Out]

(3*b*x^(5/3))/5 + (a*x^2)/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a \int x\, dx + \frac{3 b x^{\frac{5}{3}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*Integral(x, x) + 3*b*x**(5/3)/5

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Mathematica [A]  time = 0.00468519, size = 19, normalized size = 1. \[ \frac{a x^2}{2}+\frac{3}{5} b x^{5/3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^(1/3))*x,x]

[Out]

(3*b*x^(5/3))/5 + (a*x^2)/2

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Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \[{\frac{3\,b}{5}{x}^{{\frac{5}{3}}}}+{\frac{a{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))*x,x)

[Out]

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

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Maxima [A]  time = 1.41882, size = 20, normalized size = 1.05 \[ \frac{1}{10} \,{\left (5 \, a + \frac{6 \, b}{x^{\frac{1}{3}}}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(5*a + 6*b/x^(1/3))*x^2

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Fricas [A]  time = 0.21738, size = 18, normalized size = 0.95 \[ \frac{1}{2} \, a x^{2} + \frac{3}{5} \, b x^{\frac{5}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a*x^2 + 3/5*b*x^(5/3)

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Sympy [A]  time = 2.59045, size = 15, normalized size = 0.79 \[ \frac{a x^{2}}{2} + \frac{3 b x^{\frac{5}{3}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**(1/3))*x,x)

[Out]

a*x**2/2 + 3*b*x**(5/3)/5

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GIAC/XCAS [A]  time = 0.220028, size = 18, normalized size = 0.95 \[ \frac{1}{2} \, a x^{2} + \frac{3}{5} \, b x^{\frac{5}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a*x^2 + 3/5*b*x^(5/3)