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

Optimal. Leaf size=25 \[ -\frac {\left (a \left (b x^n\right )^p\right )^q}{x (1-n p q)} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, 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} \[ -\frac {\left (a \left (b x^n\right )^p\right )^q}{x (1-n p q)} \]

Antiderivative was successfully verified.

[In]

Int[(a*(b*x^n)^p)^q/x^2,x]

[Out]

-((a*(b*x^n)^p)^q/((1 - n*p*q)*x))

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

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.92 \[ \frac {\left (a \left (b x^n\right )^p\right )^q}{x (n p q-1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(b*x^n)^p)^q/((-1 + n*p*q)*x)

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fricas [A]  time = 0.56, size = 29, normalized size = 1.16 \[ \frac {e^{\left (n p q \log \relax (x) + p q \log \relax (b) + q \log \relax (a)\right )}}{{\left (n p q - 1\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((b*x^n)^p*a)^q/x^2, x)

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maple [A]  time = 0.00, size = 24, normalized size = 0.96 \[ \frac {\left (a \left (b \,x^{n}\right )^{p}\right )^{q}}{\left (n p q -1\right ) x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.80, size = 27, normalized size = 1.08 \[ \frac {a^{q} b^{p q} {\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{{\left (n p q - 1\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*(b*x^n)^p)^q/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*(b*x**n)**p)**q/x**2, x)

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