3.205 \(\int x^{-2-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx\)

Optimal. Leaf size=22 \[ -x^{-n p q-1} \left (a \left (b x^n\right )^p\right )^q \]

[Out]

-(x^(-1 - n*p*q)*(a*(b*x^n)^p)^q)

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Rubi [A]  time = 0.0743356, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -x^{-n p q-1} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

[In]  Int[x^(-2 - n*p*q)*(a*(b*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 = 8.78086, size = 19, normalized size = 0.86 \[ - \frac{x^{- n p q} \left (a \left (b x^{n}\right )^{p}\right )^{q}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-n*p*q-2)*(a*(b*x**n)**p)**q,x)

[Out]

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

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Mathematica [A]  time = 0.01136, size = 22, normalized size = 1. \[ -x^{-n p q-1} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-2 - n*p*q)*(a*(b*x^n)^p)^q,x]

[Out]

-(x^(-1 - n*p*q)*(a*(b*x^n)^p)^q)

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Maple [A]  time = 0.003, size = 23, normalized size = 1.1 \[ -{x}^{-npq-1} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-n*p*q-2)*(a*(b*x^n)^p)^q,x)

[Out]

-x^(-n*p*q-1)*(a*(b*x^n)^p)^q

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Maxima [A]  time = 1.8285, size = 18, normalized size = 0.82 \[ -\frac{a^{q}{\left (b^{p}\right )}^{q}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n)^p*a)^q*x^(-n*p*q - 2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.230161, size = 22, normalized size = 1. \[ -\frac{e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n)^p*a)^q*x^(-n*p*q - 2),x, algorithm="fricas")

[Out]

-e^(p*q*log(b) + q*log(a))/x

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-n*p*q-2)*(a*(b*x**n)**p)**q,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.333306, size = 24, normalized size = 1.09 \[ -x e^{\left (p q{\rm ln}\left (b\right ) + q{\rm ln}\left (a\right ) - 2 \,{\rm ln}\left (x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n)^p*a)^q*x^(-n*p*q - 2),x, algorithm="giac")

[Out]

-x*e^(p*q*ln(b) + q*ln(a) - 2*ln(x))