3.192 \(\int \left (a \left (b x^n\right )^p\right )^q \, dx\)

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

[Out]

(x*(a*(b*x^n)^p)^q)/(1 + n*p*q)

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Rubi [A]  time = 0.0304483, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \left (a \left (b x^n\right )^p\right )^q}{n p q+1} \]

Antiderivative was successfully verified.

[In]  Int[(a*(b*x^n)^p)^q,x]

[Out]

(x*(a*(b*x^n)^p)^q)/(1 + n*p*q)

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Rubi in Sympy [A]  time = 2.51704, size = 31, normalized size = 1.48 \[ \frac{x^{- n p q} x^{n p q + 1} \left (a \left (b x^{n}\right )^{p}\right )^{q}}{n p q + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00541827, size = 21, normalized size = 1. \[ \frac{x \left (a \left (b x^n\right )^p\right )^q}{n p q+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*(b*x^n)^p)^q,x]

[Out]

(x*(a*(b*x^n)^p)^q)/(1 + n*p*q)

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Maple [A]  time = 0.002, size = 22, normalized size = 1.1 \[{\frac{x \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{npq+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*(b*x^n)^p)^q,x)

[Out]

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

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Maxima [A]  time = 1.65177, size = 34, normalized size = 1.62 \[ \frac{a^{q}{\left (b^{p}\right )}^{q} x{\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{n p q + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^q*(b^p)^q*x*((x^n)^p)^q/(n*p*q + 1)

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Fricas [A]  time = 0.236525, size = 36, normalized size = 1.71 \[ \frac{x e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*(b*x**n)**p)**q,x)

[Out]

Integral((a*(b*x**n)**p)**q, x)

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GIAC/XCAS [A]  time = 0.238163, size = 36, normalized size = 1.71 \[ \frac{x e^{\left (n p q{\rm ln}\left (x\right ) + p q{\rm ln}\left (b\right ) + q{\rm ln}\left (a\right )\right )}}{n p q + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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