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

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

[Out]

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

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Rubi [A]  time = 0.0118048, 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 \[ x^{m+1} \log (x) \left (a x^n\right )^{-\frac{m+1}{n}} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a*x^n)^((1 + m)/n),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.81269, size = 17, normalized size = 0.77 \[ x^{m + 1} \left (a x^{n}\right )^{- \frac{m + 1}{n}} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0148402, size = 22, normalized size = 1. \[ x^{m+1} \log (x) \left (a x^n\right )^{-\frac{m+1}{n}} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/((a*x^n)^((1+m)/n)),x)

[Out]

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

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Maxima [A]  time = 1.51359, size = 23, normalized size = 1.05 \[ a^{-\frac{m}{n} - \frac{1}{n}} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^(-m/n - 1/n)*log(x)

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Fricas [A]  time = 0.238728, size = 19, normalized size = 0.86 \[ \frac{\log \left (x\right )}{a^{\frac{m + 1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)/a^((m + 1)/n)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\left (a x^{n}\right )^{\frac{m + 1}{n}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(a*x^n)^((m + 1)/n), x)