3.193 \(\int \frac{\left (a \left (b x^n\right )^p\right )^q}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0803711, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\left (a \left (b x^n\right )^p\right )^q}{n p q} \]

Antiderivative was successfully verified.

[In]  Int[(a*(b*x^n)^p)^q/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.26835, size = 14, normalized size = 0.67 \[ \frac{\left (a \left (b x^{n}\right )^{p}\right )^{q}}{n p q} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00478247, size = 21, normalized size = 1. \[ \frac{\left (a \left (b x^n\right )^p\right )^q}{n p q} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*(b*x^n)^p)^q/x,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*(b*x^n)^p)^q/x,x)

[Out]

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

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Maxima [A]  time = 1.65262, size = 34, normalized size = 1.62 \[ \frac{a^{q}{\left (b^{p}\right )}^{q}{\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{n p q} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.241137, size = 36, normalized size = 1.71 \[ \frac{e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.61234, size = 42, normalized size = 2. \[ \begin{cases} \log{\left (x \right )} & \text{for}\: q = 0 \wedge \left (n = 0 \vee q = 0\right ) \wedge \left (p = 0 \vee q = 0\right ) \\\left (a b^{p}\right )^{q} \log{\left (x \right )} & \text{for}\: n = 0 \\a^{q} \log{\left (x \right )} & \text{for}\: p = 0 \\\frac{a^{q} \left (b^{p}\right )^{q} \left (\left (x^{n}\right )^{p}\right )^{q}}{n p q} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((log(x), Eq(q, 0) & (Eq(n, 0) | Eq(q, 0)) & (Eq(p, 0) | Eq(q, 0))), ((
a*b**p)**q*log(x), Eq(n, 0)), (a**q*log(x), Eq(p, 0)), (a**q*(b**p)**q*((x**n)**
p)**q/(n*p*q), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(((b*x^n)^p*a)^q/x, x)