3.2.91 \(\int x (a (b x^n)^p)^q \, dx\) [191]

Optimal. Leaf size=23 \[ \frac {x^2 \left (a \left (b x^n\right )^p\right )^q}{2+n p q} \]

[Out]

x^2*(a*(b*x^n)^p)^q/(n*p*q+2)

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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1971, 30} \begin {gather*} \frac {x^2 \left (a \left (b x^n\right )^p\right )^q}{n p q+2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(a*(b*x^n)^p)^q)/(2 + 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 x \left (a \left (b x^n\right )^p\right )^q \, 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 {x^2 \left (a \left (b x^n\right )^p\right )^q}{2+n p q}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} \frac {x^2 \left (a \left (b x^n\right )^p\right )^q}{2+n p q} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(a*(b*x^n)^p)^q)/(2 + n*p*q)

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Maple [A]
time = 0.05, size = 24, normalized size = 1.04

method	result	size
gosper	\(\frac {x^{2} \left (a \left (b \,x^{n}\right )^{p}\right )^{q}}{n p q +2}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*(a*(b*x^n)^p)^q/(n*p*q+2)

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Maxima [A]
time = 0.38, size = 27, normalized size = 1.17 \begin {gather*} \frac {a^{q} b^{p q} x^{2} {\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{n p q + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^q*b^(p*q)*x^2*((x^n)^p)^q/(n*p*q + 2)

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Fricas [A]
time = 0.35, size = 29, normalized size = 1.26 \begin {gather*} \frac {x^{2} e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*e^(n*p*q*log(x) + p*q*log(b) + q*log(a))/(n*p*q + 2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a \left (b x^{n}\right )^{p}\right )^{q}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a*(b*x**n)**p)**q,x)

[Out]

Integral(x*(a*(b*x**n)**p)**q, x)

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Giac [A]
time = 2.09, size = 29, normalized size = 1.26 \begin {gather*} \frac {x^{2} e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*e^(n*p*q*log(x) + p*q*log(b) + q*log(a))/(n*p*q + 2)

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Mupad [B]
time = 0.99, size = 23, normalized size = 1.00 \begin {gather*} \frac {x^2\,{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{n\,p\,q+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*(a*(b*x^n)^p)^q)/(n*p*q + 2)

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