3.101 \(\int \sqrt [3]{b x^n} \, dx\)

Optimal. Leaf size=17 \[ \frac{3 x \sqrt [3]{b x^n}}{n+3} \]

[Out]

(3*x*(b*x^n)^(1/3))/(3 + n)

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Rubi [A]  time = 0.014436, antiderivative size = 17, 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 \[ \frac{3 x \sqrt [3]{b x^n}}{n+3} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^n)^(1/3),x]

[Out]

(3*x*(b*x^n)^(1/3))/(3 + n)

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Rubi in Sympy [A]  time = 2.07228, size = 24, normalized size = 1.41 \[ \frac{3 x^{- \frac{n}{3}} x^{\frac{n}{3} + 1} \sqrt [3]{b x^{n}}}{n + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00467975, size = 17, normalized size = 1. \[ \frac{3 x \sqrt [3]{b x^n}}{n+3} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^n)^(1/3),x]

[Out]

(3*x*(b*x^n)^(1/3))/(3 + n)

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Maple [A]  time = 0.002, size = 16, normalized size = 0.9 \[ 3\,{\frac{x\sqrt [3]{b{x}^{n}}}{3+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^n)^(1/3),x)

[Out]

3*x*(b*x^n)^(1/3)/(3+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)^(1/3),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)^(1/3),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)**(1/3),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.227159, size = 23, normalized size = 1.35 \[ \frac{3 \, b^{\frac{1}{3}} x e^{\left (\frac{1}{3} \, n{\rm ln}\left (x\right )\right )}}{n + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*b^(1/3)*x*e^(1/3*n*ln(x))/(n + 3)