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

Optimal. Leaf size=25 \[ \frac{1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

[Out]

x^3/(2*(a*(b*x^m)^n)^(1/(m*n)))

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Rubi [A]  time = 0.085973, 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 \[ \frac{1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

Antiderivative was successfully verified.

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

[Out]

x^3/(2*(a*(b*x^m)^n)^(1/(m*n)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0175664, size = 25, normalized size = 1. \[ \frac{1}{2} x^3 \left (a \left (b x^m\right )^n\right )^{-\frac{1}{m n}} \]

Antiderivative was successfully verified.

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

[Out]

x^3/(2*(a*(b*x^m)^n)^(1/(m*n)))

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Maple [A]  time = 0.003, size = 25, normalized size = 1. \[{\frac{{x}^{3}}{2} \left ( \left ( a \left ( b{x}^{m} \right ) ^{n} \right ) ^{{\frac{1}{mn}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^3/((a*(b*x^m)^n)^(1/m/n))

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Maxima [A]  time = 1.71287, size = 55, normalized size = 2.2 \[ \frac{1}{2} \, a^{-\frac{1}{m n}}{\left (b^{n}\right )}^{-\frac{1}{m n}} x^{3}{\left ({\left (x^{m}\right )}^{n}\right )}^{-\frac{1}{m n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^(-1/(m*n))*(b^n)^(-1/(m*n))*x^3*((x^m)^n)^(-1/(m*n))

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Fricas [A]  time = 0.236403, size = 28, normalized size = 1.12 \[ \frac{1}{2} \, x^{2} e^{\left (-\frac{n \log \left (b\right ) + \log \left (a\right )}{m n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: RecursionError

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GIAC/XCAS [A]  time = 0.276686, size = 28, normalized size = 1.12 \[ \frac{1}{2} \, x^{2} e^{\left (-\frac{n{\rm ln}\left (b\right ) +{\rm ln}\left (a\right )}{m n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2*e^(-(n*ln(b) + ln(a))/(m*n))