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

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

[Out]

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

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Rubi [A]  time = 0.0757675, antiderivative size = 21, 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 \[ \log (x) x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n*p*q)*(a*(b*x^n)^p)^q,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.69296, size = 19, normalized size = 0.9 \[ x^{- n p q} \left (a \left (b x^{n}\right )^{p}\right )^{q} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0114324, size = 21, normalized size = 1. \[ \log (x) x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.252, size = 0, normalized size = 0. \[ \int{x}^{-npq-1} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-n*p*q-1)*(a*(b*x^n)^p)^q,x)

[Out]

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

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Maxima [A]  time = 1.83342, size = 15, normalized size = 0.71 \[ a^{q}{\left (b^{p}\right )}^{q} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.233747, size = 19, normalized size = 0.9 \[ e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^(p*q*log(b) + q*log(a))*log(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-1)*(a*(b*x**n)**p)**q,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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