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

Optimal. Leaf size=24 \[ \frac{1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \]

[Out]

(x^(2 - n*p*q)*(a*(b*x^n)^p)^q)/2

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Rubi [A]  time = 0.0750715, antiderivative size = 24, 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 \[ \frac{1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

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

[Out]

(x^(2 - n*p*q)*(a*(b*x^n)^p)^q)/2

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

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*Integral(x, x)

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Mathematica [A]  time = 0.00752536, size = 24, normalized size = 1. \[ \frac{1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

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

[Out]

(x^(2 - n*p*q)*(a*(b*x^n)^p)^q)/2

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Maple [A]  time = 0.002, size = 23, normalized size = 1. \[{\frac{{x}^{-npq+2} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{2}} \]

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]

1/2*x^(-n*p*q+2)*(a*(b*x^n)^p)^q

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Maxima [A]  time = 1.77212, size = 18, normalized size = 0.75 \[ \frac{1}{2} \, a^{q}{\left (b^{p}\right )}^{q} x^{2} \]

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]

1/2*a^q*(b^p)^q*x^2

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Fricas [A]  time = 0.229152, size = 22, normalized size = 0.92 \[ \frac{1}{2} \, x^{2} e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x^2*e^(p*q*log(b) + q*log(a))

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

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]

Integral(x**(-n*p*q + 1)*(a*(b*x**n)**p)**q, x)

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GIAC/XCAS [A]  time = 0.280279, size = 22, normalized size = 0.92 \[ \frac{1}{2} \, x e^{\left (p q{\rm ln}\left (b\right ) + q{\rm ln}\left (a\right ) +{\rm ln}\left (x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x*e^(p*q*ln(b) + q*ln(a) + ln(x))