3.3.5 \(\int x^{-2-n p q} (a (b x^n)^p)^q \, dx\) [205]

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

[Out]

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

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Rubi [A]
time = 0.01, 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 = {1971, 30} \begin {gather*} -x^{-n p q-1} \left (a \left (b x^n\right )^p\right )^q \end {gather*}

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 1971

Int[(u_.)*((c_.)*((d_.)*((a_.) + (b_.)*(x_))^(n_))^(q_))^(p_), x_Symbol] :> Dist[(c*(d*(a + b*x)^n)^q)^p/(a +
b*x)^(n*p*q), Int[u*(a + b*x)^(n*p*q), x], x] /; FreeQ[{a, b, c, d, n, q, p}, x] &&  !IntegerQ[q] &&  !Integer
Q[p]

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 = 0.00, size = 22, normalized size = 1.00 \begin {gather*} -x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \end {gather*}

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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Maple [A]
time = 0.20, size = 23, normalized size = 1.05

method	result	size
gosper	\(-x^{-n p q -1} \left (a \left (b \,x^{n}\right )^{p}\right )^{q}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 0.35, size = 16, normalized size = 0.73 \begin {gather*} -\frac {e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )}}{x} \end {gather*}

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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Sympy [A]
time = 6.25, size = 19, normalized size = 0.86 \begin {gather*} - \frac {x^{- n p q} \left (a \left (b x^{n}\right )^{p}\right )^{q}}{x} \end {gather*}

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

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Giac [A]
time = 1.45, size = 18, normalized size = 0.82 \begin {gather*} -x e^{\left (p q \log \left (b\right ) + q \log \left (a\right ) - 2 \, \log \left (x\right )\right )} \end {gather*}

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

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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