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

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Rubi [A]  time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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]

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*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \[ -x^{-n p q-1} \left (a \left (b x^n\right )^p\right )^q \]

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)

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fricas [A]  time = 0.76, size = 16, normalized size = 0.73 \[ -\frac {e^{\left (p q \log \relax (b) + q \log \relax (a)\right )}}{x} \]

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

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giac [A]  time = 0.28, size = 18, normalized size = 0.82 \[ -x e^{\left (p q \log \relax (b) + q \log \relax (a) - 2 \, \log \relax (x)\right )} \]

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))

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maple [A]  time = 0.00, size = 23, normalized size = 1.05 \[ -x^{-n p q -1} \left (a \left (b \,x^{n}\right )^{p}\right )^{q} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\left (b x^{n}\right )^{p} a\right )^{q} x^{-n p q - 2}\,{d x} \]

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)

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mupad [B]  time = 1.01, size = 23, normalized size = 1.05 \[ -\frac {{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{x^{n\,p\,q+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*(b*x^n)^p)^q/x^(n*p*q + 2),x)

[Out]

-(a*(b*x^n)^p)^q/x^(n*p*q + 1)

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sympy [A]  time = 93.32, size = 24, normalized size = 1.09 \[ - \frac {a^{q} x^{- n p q} \left (b^{p}\right )^{q} \left (\left (x^{n}\right )^{p}\right )^{q}}{x} \]

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

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