3.116 \(\int \frac{1}{\sqrt [3]{b x^n}} \, dx\)

Optimal. Leaf size=19 \[ \frac{3 x}{(3-n) \sqrt [3]{b x^n}} \]

[Out]

(3*x)/((3 - n)*(b*x^n)^(1/3))

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Rubi [A]  time = 0.0142028, 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 \[ \frac{3 x}{(3-n) \sqrt [3]{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^n)^(-1/3),x]

[Out]

(3*x)/((3 - n)*(b*x^n)^(1/3))

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Rubi in Sympy [A]  time = 2.83268, size = 27, normalized size = 1.42 \[ \frac{3 x^{- \frac{2 n}{3}} x^{- \frac{n}{3} + 1} \left (b x^{n}\right )^{\frac{2}{3}}}{b \left (- n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0082226, size = 17, normalized size = 0.89 \[ -\frac{3 x}{(n-3) \sqrt [3]{b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^n)^(-1/3),x]

[Out]

(-3*x)/((-3 + n)*(b*x^n)^(1/3))

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Maple [A]  time = 0.002, size = 16, normalized size = 0.8 \[ -3\,{\frac{x}{ \left ( -3+n \right ) \sqrt [3]{b{x}^{n}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^n)^(1/3),x)

[Out]

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

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x^{n}\right )^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n)^(-1/3), x)