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

Optimal. Leaf size=21 \[ \frac{x \left (a x^n\right )^{-1/n} (c x)^m}{m} \]

[Out]

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

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Rubi [A]  time = 0.0146271, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{x \left (a x^n\right )^{-1/n} (c x)^m}{m} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^m/(a*x^n)^n^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.08082, size = 15, normalized size = 0.71 \[ \frac{x \left (a x^{n}\right )^{- \frac{1}{n}} \left (c x\right )^{m}}{m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00497446, size = 21, normalized size = 1. \[ \frac{x \left (a x^n\right )^{-1/n} (c x)^m}{m} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.002, size = 22, normalized size = 1.1 \[{\frac{x \left ( cx \right ) ^{m}}{m\sqrt [n]{a{x}^{n}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^m/((a*x^n)^(1/n)),x)

[Out]

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

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Maxima [A]  time = 1.4973, size = 41, normalized size = 1.95 \[ \frac{a^{-\frac{1}{n}} c^{m} x e^{\left (m \log \left (x\right ) - \frac{\log \left (x^{n}\right )}{n}\right )}}{m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.235837, size = 28, normalized size = 1.33 \[ \frac{e^{\left (m \log \left (c\right ) + m \log \left (x\right )\right )}}{a^{\left (\frac{1}{n}\right )} m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^(m*log(c) + m*log(x))/(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((c*x)**m/((a*x**n)**(1/n)),x)

[Out]

Exception raised: RecursionError

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GIAC/XCAS [A]  time = 0.231642, size = 28, normalized size = 1.33 \[ \frac{e^{\left (m{\rm ln}\left (c\right ) + m{\rm ln}\left (x\right ) - \frac{{\rm ln}\left (a\right )}{n}\right )}}{m} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^(m*ln(c) + m*ln(x) - ln(a)/n)/m