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

Optimal. Leaf size=26 \[ \frac{x^{m+1} \left (a \left (b x^n\right )^p\right )^q}{m+n p q+1} \]

[Out]

(x^(1 + m)*(a*(b*x^n)^p)^q)/(1 + m + n*p*q)

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Rubi [A]  time = 0.0451479, antiderivative size = 26, 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} \[ \frac{x^{m+1} \left (a \left (b x^n\right )^p\right )^q}{m+n p q+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0078345, size = 26, normalized size = 1. \[ \frac{x^{m+1} \left (a \left (b x^n\right )^p\right )^q}{m+n p q+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*(a*(b*x^n)^p)^q)/(1 + m + n*p*q)

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Maple [A]  time = 0.003, size = 27, normalized size = 1. \begin{align*}{\frac{{x}^{1+m} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{npq+m+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(1+m)*(a*(b*x^n)^p)^q/(n*p*q+m+1)

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Maxima [A]  time = 1.42313, size = 45, normalized size = 1.73 \begin{align*} \frac{a^{q}{\left (b^{p}\right )}^{q} x e^{\left (m \log \left (x\right ) + q \log \left ({\left (x^{n}\right )}^{p}\right )\right )}}{n p q + m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^q*(b^p)^q*x*e^(m*log(x) + q*log((x^n)^p))/(n*p*q + m + 1)

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Fricas [A]  time = 1.77252, size = 86, normalized size = 3.31 \begin{align*} \frac{x x^{m} e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q + m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*x^m*e^(n*p*q*log(x) + p*q*log(b) + q*log(a))/(n*p*q + m + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \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**m*(a*(b*x**n)**p)**q,x)

[Out]

Integral(x**m*(a*(b*x**n)**p)**q, x)

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Giac [A]  time = 1.12154, size = 42, normalized size = 1.62 \begin{align*} \frac{x x^{m} e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q + m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*x^m*e^(n*p*q*log(x) + p*q*log(b) + q*log(a))/(n*p*q + m + 1)