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

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 = 0.044548, 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, Rules used = {6679, 30} \[ \frac{\left (a \left (b x^n\right )^p\right )^q}{n p q} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6679

Int[(u_.)*((c_.)*((d_.)*((a_.) + (b_.)*(x_))^(n_))^(p_))^(q_), x_Symbol] :> Dist[(c*(d*(a + b*x)^n)^p)^q/(a +
b*x)^(n*p*q), Int[u*(a + b*x)^(n*p*q), x], x] /; FreeQ[{a, b, c, d, n, p, q}, x] &&  !IntegerQ[p] &&  !Integer
Q[q]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a \left (b x^n\right )^p\right )^q}{x} \, dx &=\left (x^{-n p q} \left (a \left (b x^n\right )^p\right )^q\right ) \int x^{-1+n p q} \, dx\\ &=\frac{\left (a \left (b x^n\right )^p\right )^q}{n p q}\\ \end{align*}

Mathematica [A]  time = 0.0021466, 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 \begin{align*}{\frac{ \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{npq}} \end{align*}

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 = 1.37418, size = 34, normalized size = 1.62 \begin{align*} \frac{a^{q}{\left (b^{p}\right )}^{q}{\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{n p q} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*(b*x^n)^p)^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 = 1.79109, size = 68, normalized size = 3.24 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*(b*x^n)^p)^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 = 0.771194, size = 42, normalized size = 2. \begin{align*} \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} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (b x^{n}\right )^{p} a\right )^{q}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((b*x^n)^p*a)^q/x, x)