3.202 \(\int x^{1-n p q} (a (b x^n)^p)^q \, dx\)

Optimal. Leaf size=24 \[ \frac{1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \]

[Out]

(x^(2 - n*p*q)*(a*(b*x^n)^p)^q)/2

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Rubi [A]  time = 0.0421745, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {6679, 30} \[ \frac{1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

[In]

Int[x^(1 - n*p*q)*(a*(b*x^n)^p)^q,x]

[Out]

(x^(2 - n*p*q)*(a*(b*x^n)^p)^q)/2

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^{1-n p q} \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 \, dx\\ &=\frac{1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q\\ \end{align*}

Mathematica [A]  time = 0.0037881, size = 24, normalized size = 1. \[ \frac{1}{2} x^{2-n p q} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

[In]

Integrate[x^(1 - n*p*q)*(a*(b*x^n)^p)^q,x]

[Out]

(x^(2 - n*p*q)*(a*(b*x^n)^p)^q)/2

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Maple [A]  time = 0.001, size = 23, normalized size = 1. \begin{align*}{\frac{{x}^{-npq+2} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-n*p*q+1)*(a*(b*x^n)^p)^q,x)

[Out]

1/2*x^(-n*p*q+2)*(a*(b*x^n)^p)^q

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (b x^{n}\right )^{p} a\right )^{q} x^{-n p q + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-n*p*q+1)*(a*(b*x^n)^p)^q,x, algorithm="maxima")

[Out]

integrate(((b*x^n)^p*a)^q*x^(-n*p*q + 1), x)

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Fricas [A]  time = 1.71207, size = 47, normalized size = 1.96 \begin{align*} \frac{1}{2} \, x^{2} e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-n*p*q+1)*(a*(b*x^n)^p)^q,x, algorithm="fricas")

[Out]

1/2*x^2*e^(p*q*log(b) + q*log(a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{- n p q + 1} \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**(-n*p*q+1)*(a*(b*x**n)**p)**q,x)

[Out]

Integral(x**(-n*p*q + 1)*(a*(b*x**n)**p)**q, x)

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Giac [A]  time = 1.15266, size = 22, normalized size = 0.92 \begin{align*} \frac{1}{2} \, x e^{\left (p q \log \left (b\right ) + q \log \left (a\right ) + \log \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-n*p*q+1)*(a*(b*x^n)^p)^q,x, algorithm="giac")

[Out]

1/2*x*e^(p*q*log(b) + q*log(a) + log(x))