3.124 \(\int \frac{1}{\left (b x^n\right )^{2/3}} \, dx\)

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

[Out]

(3*x)/((3 - 2*n)*(b*x^n)^(2/3))

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Rubi [A]  time = 0.0159956, 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 \[ \frac{3 x}{(3-2 n) \left (b x^n\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^n)^(-2/3),x]

[Out]

(3*x)/((3 - 2*n)*(b*x^n)^(2/3))

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Rubi in Sympy [A]  time = 2.76107, size = 29, normalized size = 1.53 \[ \frac{3 x^{- \frac{n}{3}} x^{- \frac{2 n}{3} + 1} \sqrt [3]{b x^{n}}}{b \left (- 2 n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**n)**(2/3),x)

[Out]

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

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Mathematica [A]  time = 0.00954605, size = 20, normalized size = 1.05 \[ \frac{x}{\left (1-\frac{2 n}{3}\right ) \left (b x^n\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^n)^(-2/3),x]

[Out]

x/((1 - (2*n)/3)*(b*x^n)^(2/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^n)^(2/3),x)

[Out]

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

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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)^(-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]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x^{n}\right )^{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**n)**(2/3),x)

[Out]

Integral((b*x**n)**(-2/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (b x^{n}\right )^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n)^(-2/3), x)