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

Optimal. Leaf size=22 \[ x \log (x) \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

[Out]

(x*Log[x])/(a*(b*x^m)^n)^(1/(m*n))

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Rubi [A]  time = 0.02475, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ x \log (x) \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Log[x])/(a*(b*x^m)^n)^(1/(m*n))

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Rubi in Sympy [A]  time = 2.15424, size = 17, normalized size = 0.77 \[ x \left (a \left (b x^{m}\right )^{n}\right )^{- \frac{1}{m n}} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0029464, size = 22, normalized size = 1. \[ x \log (x) \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Log[x])/(a*(b*x^m)^n)^(1/(m*n))

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int \left ( \left ( a \left ( b{x}^{m} \right ) ^{n} \right ) ^{{\frac{1}{mn}}} \right ) ^{-1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.81124, size = 34, normalized size = 1.55 \[ a^{-\frac{1}{m n}}{\left (b^{n}\right )}^{-\frac{1}{m n}} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232732, size = 26, normalized size = 1.18 \[ e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^(-(n*log(b) + log(a))/(m*n))*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.253465, size = 26, normalized size = 1.18 \[ e^{\left (-\frac{n{\rm ln}\left (b\right ) +{\rm ln}\left (a\right )}{m n}\right )}{\rm ln}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^(-(n*ln(b) + ln(a))/(m*n))*ln(x)