3.180 \(\int x^m \left (a x^n\right )^{-1/n} \, dx\)

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

[Out]

x^(1 + m)/(m*(a*x^n)^n^(-1))

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Rubi [A]  time = 0.0112106, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^{m+1} \left (a x^n\right )^{-1/n}}{m} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a*x^n)^n^(-1),x]

[Out]

x^(1 + m)/(m*(a*x^n)^n^(-1))

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Rubi in Sympy [A]  time = 2.74113, size = 14, normalized size = 0.7 \[ \frac{x x^{m} \left (a x^{n}\right )^{- \frac{1}{n}}}{m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00847059, size = 20, normalized size = 1. \[ \frac{x^{m+1} \left (a x^n\right )^{-1/n}}{m} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(a*x^n)^n^(-1),x]

[Out]

x^(1 + m)/(m*(a*x^n)^n^(-1))

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Maple [A]  time = 0.002, size = 21, normalized size = 1.1 \[{\frac{{x}^{1+m}}{m\sqrt [n]{a{x}^{n}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/((a*x^n)^(1/n)),x)

[Out]

x^(1+m)/m/((a*x^n)^(1/n))

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Maxima [A]  time = 1.46318, size = 36, normalized size = 1.8 \[ \frac{a^{-\frac{1}{n}} x e^{\left (m \log \left (x\right ) - \frac{\log \left (x^{n}\right )}{n}\right )}}{m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.231711, size = 19, normalized size = 0.95 \[ \frac{x^{m}}{a^{\left (\frac{1}{n}\right )} m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: RecursionError

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GIAC/XCAS [A]  time = 0.224273, size = 23, normalized size = 1.15 \[ \frac{e^{\left (m{\rm ln}\left (x\right ) - \frac{{\rm ln}\left (a\right )}{n}\right )}}{m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^(m*ln(x) - ln(a)/n)/m