3.199 \(\int \frac{\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}}{x} \, dx\)

Optimal. Leaf size=20 \[ -\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

[Out]

-(a*(b*x^m)^n)^(-(1/(m*n)))

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Rubi [A]  time = 0.0772868, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a*(b*x^m)^n)^(1/(m*n))),x]

[Out]

-(a*(b*x^m)^n)^(-(1/(m*n)))

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Rubi in Sympy [A]  time = 6.5018, size = 15, normalized size = 0.75 \[ - \left (a \left (b x^{m}\right )^{n}\right )^{- \frac{1}{m n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*(b*x**m)**n)**(-1/(m*n))

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Mathematica [A]  time = 0.00405322, size = 20, normalized size = 1. \[ -\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a*(b*x^m)^n)^(1/(m*n))),x]

[Out]

-(a*(b*x^m)^n)^(-(1/(m*n)))

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Maple [A]  time = 0.003, size = 22, normalized size = 1.1 \[ - \left ( \left ( a \left ( b{x}^{m} \right ) ^{n} \right ) ^{{\frac{1}{mn}}} \right ) ^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/((a*(b*x^m)^n)^(1/m/n)),x)

[Out]

-1/((a*(b*x^m)^n)^(1/m/n))

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Maxima [A]  time = 1.7097, size = 51, normalized size = 2.55 \[ -a^{-\frac{1}{m n}}{\left (b^{n}\right )}^{-\frac{1}{m n}}{\left ({\left (x^{m}\right )}^{n}\right )}^{-\frac{1}{m n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a^(-1/(m*n))*(b^n)^(-1/(m*n))*((x^m)^n)^(-1/(m*n))

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Fricas [A]  time = 0.231991, size = 28, normalized size = 1.4 \[ -\frac{e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e^(-(n*log(b) + log(a))/(m*n))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/((a*(b*x**m)**n)**(1/m/n)),x)

[Out]

Exception raised: RecursionError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(((b*x^m)^n*a)^(1/(m*n))*x), x)