3.205 \(\int x^{-2-n p q} (a (b x^n)^p)^q \, dx\)

Optimal. Leaf size=22 \[ -x^{-n p q-1} \left (a \left (b x^n\right )^p\right )^q \]

[Out]

-(x^(-1 - n*p*q)*(a*(b*x^n)^p)^q)

________________________________________________________________________________________

Rubi [A]  time = 0.0420725, antiderivative size = 22, 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} \[ -x^{-n p q-1} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

[In]

Int[x^(-2 - n*p*q)*(a*(b*x^n)^p)^q,x]

[Out]

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

Mathematica [A]  time = 0.0050027, size = 22, normalized size = 1. \[ -x^{-n p q-1} \left (a \left (b x^n\right )^p\right )^q \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-2 - n*p*q)*(a*(b*x^n)^p)^q,x]

[Out]

-(x^(-1 - n*p*q)*(a*(b*x^n)^p)^q)

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 23, normalized size = 1.1 \begin{align*} -{x}^{-npq-1} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-n*p*q-2)*(a*(b*x^n)^p)^q,x)

[Out]

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

________________________________________________________________________________________

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 - 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-n*p*q-2)*(a*(b*x^n)^p)^q,x, algorithm="maxima")

[Out]

integrate(((b*x^n)^p*a)^q*x^(-n*p*q - 2), x)

________________________________________________________________________________________

Fricas [A]  time = 1.81456, size = 41, normalized size = 1.86 \begin{align*} -\frac{e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-n*p*q-2)*(a*(b*x^n)^p)^q,x, algorithm="fricas")

[Out]

-e^(p*q*log(b) + q*log(a))/x

________________________________________________________________________________________

Sympy [A]  time = 145.066, size = 24, normalized size = 1.09 \begin{align*} - \frac{a^{q} x^{- n p q} \left (b^{p}\right )^{q} \left (\left (x^{n}\right )^{p}\right )^{q}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-n*p*q-2)*(a*(b*x**n)**p)**q,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.17026, size = 24, normalized size = 1.09 \begin{align*} -x e^{\left (p q \log \left (b\right ) + q \log \left (a\right ) - 2 \, \log \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-n*p*q-2)*(a*(b*x^n)^p)^q,x, algorithm="giac")

[Out]

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