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

Optimal. Leaf size=24 \[ \frac{1}{3} x^{3-n p q} \left (a \left (b x^n\right )^p\right )^q \]

[Out]

(x^(3 - n*p*q)*(a*(b*x^n)^p)^q)/3

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Rubi [A]  time = 0.0733942, 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 \[ \frac{1}{3} x^{3-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

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

[Out]

(x^(3 - n*p*q)*(a*(b*x^n)^p)^q)/3

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Rubi in Sympy [A]  time = 8.85048, size = 20, normalized size = 0.83 \[ \frac{x^{3} x^{- n p q} \left (a \left (b x^{n}\right )^{p}\right )^{q}}{3} \]

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**3*x**(-n*p*q)*(a*(b*x**n)**p)**q/3

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Mathematica [A]  time = 0.00978924, size = 24, normalized size = 1. \[ \frac{1}{3} x^{3-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

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

[Out]

(x^(3 - n*p*q)*(a*(b*x^n)^p)^q)/3

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Maple [A]  time = 0.003, size = 23, normalized size = 1. \[{\frac{{x}^{-npq+3} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{3}} \]

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]

1/3*x^(-n*p*q+3)*(a*(b*x^n)^p)^q

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Maxima [A]  time = 1.84018, size = 18, normalized size = 0.75 \[ \frac{1}{3} \, a^{q}{\left (b^{p}\right )}^{q} x^{3} \]

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]

1/3*a^q*(b^p)^q*x^3

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Fricas [A]  time = 0.229507, size = 22, normalized size = 0.92 \[ \frac{1}{3} \, x^{3} e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3*e^(p*q*log(b) + q*log(a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{- n p q + 2} \left (a \left (b x^{n}\right )^{p}\right )^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**(-n*p*q + 2)*(a*(b*x**n)**p)**q, x)

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GIAC/XCAS [A]  time = 0.25519, size = 24, normalized size = 1. \[ \frac{1}{3} \, x e^{\left (p q{\rm ln}\left (b\right ) + q{\rm ln}\left (a\right ) + 2 \,{\rm ln}\left (x\right )\right )} \]

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]

1/3*x*e^(p*q*ln(b) + q*ln(a) + 2*ln(x))