3.188 \(\int x^{-1-n p} \left (a x^n\right )^p \, dx\)

Optimal. Leaf size=16 \[ \log (x) x^{-n p} \left (a x^n\right )^p \]

[Out]

((a*x^n)^p*Log[x])/x^(n*p)

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Rubi [A]  time = 0.00998475, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \log (x) x^{-n p} \left (a x^n\right )^p \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n*p)*(a*x^n)^p,x]

[Out]

((a*x^n)^p*Log[x])/x^(n*p)

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Rubi in Sympy [A]  time = 2.36821, size = 14, normalized size = 0.88 \[ x^{- n p} \left (a x^{n}\right )^{p} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-n*p-1)*(a*x**n)**p,x)

[Out]

x**(-n*p)*(a*x**n)**p*log(x)

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Mathematica [A]  time = 0.0082386, size = 16, normalized size = 1. \[ \log (x) x^{-n p} \left (a x^n\right )^p \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - n*p)*(a*x^n)^p,x]

[Out]

((a*x^n)^p*Log[x])/x^(n*p)

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int{x}^{-np-1} \left ( a{x}^{n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-n*p-1)*(a*x^n)^p,x)

[Out]

int(x^(-n*p-1)*(a*x^n)^p,x)

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Maxima [A]  time = 1.50487, size = 8, normalized size = 0.5 \[ a^{p} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^p*log(x)

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Fricas [A]  time = 0.226825, size = 8, normalized size = 0.5 \[ a^{p} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^p*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-n*p-1)*(a*x**n)**p,x)

[Out]

Integral(x**(-n*p - 1)*(a*x**n)**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((a*x^n)^p*x^(-n*p - 1), x)