3.2.93 \(\int \frac {(a (b x^n)^p)^q}{x} \, dx\) [193]

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 = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 = {1971, 30} \begin {gather*} \frac {\left (a \left (b x^n\right )^p\right )^q}{n p q} \end {gather*}

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 30

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

Rule 1971

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

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*}

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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} \frac {\left (a \left (b x^n\right )^p\right )^q}{n p q} \end {gather*}

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 = 0.05, size = 22, normalized size = 1.05

method	result	size
gosper	\(\frac {\left (a \left (b \,x^{n}\right )^{p}\right )^{q}}{n p q}\)	\(22\)
derivativedivides	\(\frac {\left (a \left (b \,x^{n}\right )^{p}\right )^{q}}{n p q}\)	\(22\)
default	\(\frac {\left (a \left (b \,x^{n}\right )^{p}\right )^{q}}{n p q}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*(b*x^n)^p)^q/x,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.38, size = 25, normalized size = 1.19 \begin {gather*} \frac {a^{q} b^{p q} {\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{n p q} \end {gather*}

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 = 0.37, size = 27, normalized size = 1.29 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (14) = 28\).
time = 0.17, size = 37, normalized size = 1.76 \begin {gather*} \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 {\left (a \left (b x^{n}\right )^{p}\right )^{q}}{n p q} & \text {otherwise} \end {cases} \end {gather*}

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*(b*x**n)**p)**q/(n*p*q), True))

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Giac [A]
time = 1.87, size = 21, normalized size = 1.00 \begin {gather*} \frac {\left (\left (b x^{n}\right )^{p} a\right )^{q}}{n p q} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*(b*x^n)^p)^q/x, x)

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