3.203 \(\int x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx\)

Optimal. Leaf size=21 \[ x^{1-n p q} \left (a \left (b x^n\right )^p\right )^q \]

[Out]

x^(1 - n*p*q)*(a*(b*x^n)^p)^q

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Rubi [A]  time = 0.0748248, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ x^{1-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

[In]  Int[(a*(b*x^n)^p)^q/x^(n*p*q),x]

[Out]

x^(1 - n*p*q)*(a*(b*x^n)^p)^q

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Rubi in Sympy [A]  time = 8.17677, size = 17, normalized size = 0.81 \[ x x^{- n p q} \left (a \left (b x^{n}\right )^{p}\right )^{q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*(b*x**n)**p)**q/(x**(n*p*q)),x)

[Out]

x*x**(-n*p*q)*(a*(b*x**n)**p)**q

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Mathematica [A]  time = 0.00583489, size = 21, normalized size = 1. \[ x^{1-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

[In]  Integrate[(a*(b*x^n)^p)^q/x^(n*p*q),x]

[Out]

x^(1 - n*p*q)*(a*(b*x^n)^p)^q

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Maple [F]  time = 0.29, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{{x}^{npq}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*(b*x^n)^p)^q/(x^(n*p*q)),x)

[Out]

int((a*(b*x^n)^p)^q/(x^(n*p*q)),x)

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Maxima [A]  time = 1.74871, size = 14, normalized size = 0.67 \[ a^{q}{\left (b^{p}\right )}^{q} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n)^p*a)^q/x^(n*p*q),x, algorithm="maxima")

[Out]

a^q*(b^p)^q*x

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Fricas [A]  time = 0.228433, size = 18, normalized size = 0.86 \[ x e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n)^p*a)^q/x^(n*p*q),x, algorithm="fricas")

[Out]

x*e^(p*q*log(b) + q*log(a))

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Sympy [A]  time = 10.0414, size = 22, normalized size = 1.05 \[ a^{q} x x^{- n p q} \left (b^{p}\right )^{q} \left (\left (x^{n}\right )^{p}\right )^{q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*(b*x**n)**p)**q/(x**(n*p*q)),x)

[Out]

a**q*x*x**(-n*p*q)*(b**p)**q*((x**n)**p)**q

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GIAC/XCAS [A]  time = 0.306185, size = 18, normalized size = 0.86 \[ x e^{\left (p q{\rm ln}\left (b\right ) + q{\rm ln}\left (a\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n)^p*a)^q/x^(n*p*q),x, algorithm="giac")

[Out]

x*e^(p*q*ln(b) + q*ln(a))