3.191 \(\int x (a (b x^n)^p)^q \, dx\)

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

[Out]

(x^2*(a*(b*x^n)^p)^q)/(2 + n*p*q)

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Rubi [A]  time = 0.028609, antiderivative size = 23, normalized size of antiderivative = 1., 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 = {6679, 30} \[ \frac{x^2 \left (a \left (b x^n\right )^p\right )^q}{n p q+2} \]

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 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 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*}

Mathematica [A]  time = 0.0037556, size = 23, normalized size = 1. \[ \frac{x^2 \left (a \left (b x^n\right )^p\right )^q}{n p q+2} \]

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 = 0.001, size = 24, normalized size = 1. \begin{align*}{\frac{{x}^{2} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{npq+2}} \end{align*}

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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Maxima [A]  time = 1.39744, size = 36, normalized size = 1.57 \begin{align*} \frac{a^{q}{\left (b^{p}\right )}^{q} x^{2}{\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{n p q + 2} \end{align*}

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 = 1.77177, size = 78, normalized size = 3.39 \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int x \left (a \left (b x^{n}\right )^{p}\right )^{q}\, dx \end{align*}

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 = 1.16318, size = 39, normalized size = 1.7 \begin{align*} \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{align*}

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)