3.109 \(\int \left (b x^n\right )^{2/3} \, dx\)

Optimal. Leaf size=19 \[ \frac{3 x \left (b x^n\right )^{2/3}}{2 n+3} \]

[Out]

(3*x*(b*x^n)^(2/3))/(3 + 2*n)

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Rubi [A]  time = 0.0150834, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{3 x \left (b x^n\right )^{2/3}}{2 n+3} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^n)^(2/3),x]

[Out]

(3*x*(b*x^n)^(2/3))/(3 + 2*n)

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Rubi in Sympy [A]  time = 2.25012, size = 29, normalized size = 1.53 \[ \frac{3 x^{- \frac{2 n}{3}} x^{\frac{2 n}{3} + 1} \left (b x^{n}\right )^{\frac{2}{3}}}{2 n + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**n)**(2/3),x)

[Out]

3*x**(-2*n/3)*x**(2*n/3 + 1)*(b*x**n)**(2/3)/(2*n + 3)

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Mathematica [A]  time = 0.00462759, size = 20, normalized size = 1.05 \[ \frac{x \left (b x^n\right )^{2/3}}{\frac{2 n}{3}+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^n)^(2/3),x]

[Out]

(x*(b*x^n)^(2/3))/(1 + (2*n)/3)

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Maple [A]  time = 0.002, size = 18, normalized size = 1. \[ 3\,{\frac{x \left ( b{x}^{n} \right ) ^{2/3}}{3+2\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^n)^(2/3),x)

[Out]

3*x*(b*x^n)^(2/3)/(3+2*n)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n)^(2/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n)^(2/3),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**n)**(2/3),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.247843, size = 26, normalized size = 1.37 \[ \frac{3 \, b^{\frac{2}{3}} x e^{\left (\frac{2}{3} \, n{\rm ln}\left (x\right )\right )}}{2 \, n + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n)^(2/3),x, algorithm="giac")

[Out]

3*b^(2/3)*x*e^(2/3*n*ln(x))/(2*n + 3)