3.200 \(\int \frac{\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}}{x^2} \, dx\)

Optimal. Leaf size=25 \[ -\frac{\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}}{2 x} \]

[Out]

-1/(2*x*(a*(b*x^m)^n)^(1/(m*n)))

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Rubi [A]  time = 0.0763703, antiderivative size = 25, 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 \[ -\frac{\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}}{2 x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a*(b*x^m)^n)^(1/(m*n))),x]

[Out]

-1/(2*x*(a*(b*x^m)^n)^(1/(m*n)))

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Rubi in Sympy [A]  time = 6.56648, size = 19, normalized size = 0.76 \[ - \frac{\left (a \left (b x^{m}\right )^{n}\right )^{- \frac{1}{m n}}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*(b*x**m)**n)**(-1/(m*n))/(2*x)

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Mathematica [A]  time = 0.00662301, size = 25, normalized size = 1. \[ -\frac{\left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}}}{2 x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a*(b*x^m)^n)^(1/(m*n))),x]

[Out]

-1/(2*x*(a*(b*x^m)^n)^(1/(m*n)))

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Maple [A]  time = 0.003, size = 25, normalized size = 1. \[ -{\frac{1}{2\,x} \left ( \left ( a \left ( b{x}^{m} \right ) ^{n} \right ) ^{{\frac{1}{mn}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/((a*(b*x^m)^n)^(1/m/n)),x)

[Out]

-1/2/x/((a*(b*x^m)^n)^(1/m/n))

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Maxima [A]  time = 1.90162, size = 36, normalized size = 1.44 \[ -\frac{a^{-\frac{1}{m n}}{\left (b^{n}\right )}^{-\frac{1}{m n}}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((b*x^m)^n*a)^(1/(m*n))*x^2),x, algorithm="maxima")

[Out]

-1/2*a^(-1/(m*n))*(b^n)^(-1/(m*n))/x^2

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Fricas [A]  time = 0.235276, size = 28, normalized size = 1.12 \[ -\frac{e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((b*x^m)^n*a)^(1/(m*n))*x^2),x, algorithm="fricas")

[Out]

-1/2*e^(-(n*log(b) + log(a))/(m*n))/x^2

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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**2/((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. \[ \int \frac{1}{\left (\left (b x^{m}\right )^{n} a\right )^{\frac{1}{m n}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((b*x^m)^n*a)^(1/(m*n))*x^2),x, algorithm="giac")

[Out]

integrate(1/(((b*x^m)^n*a)^(1/(m*n))*x^2), x)