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/((1 - n*p*q)*x))

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Rubi [A]  time = 0.0452928, antiderivative size = 25, normalized size of antiderivative = 1., 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 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 \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*}

Mathematica [A]  time = 0.0049682, 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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Maple [A]  time = 0.001, size = 24, normalized size = 1. \begin{align*}{\frac{ \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{x \left ( npq-1 \right ) }} \end{align*}

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 = 1.39295, size = 36, normalized size = 1.44 \begin{align*} \frac{a^{q}{\left (b^{p}\right )}^{q}{\left ({\left (x^{n}\right )}^{p}\right )}^{q}}{{\left (n p q - 1\right )} x} \end{align*}

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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Fricas [A]  time = 1.85115, size = 78, normalized size = 3.12 \begin{align*} \frac{e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{{\left (n p q - 1\right )} x} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a \left (b x^{n}\right )^{p}\right )^{q}}{x^{2}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (b x^{n}\right )^{p} a\right )^{q}}{x^{2}}\,{d x} \end{align*}

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)