\(\int x^{-1-n p q} (a (b x^n)^p)^q \, dx\) [204]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \log (x) \]

[Out]

(a*(b*x^n)^p)^q*ln(x)/(x^(n*p*q))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1971, 29} \[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\log (x) x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \]

[In]

Int[x^(-1 - n*p*q)*(a*(b*x^n)^p)^q,x]

[Out]

((a*(b*x^n)^p)^q*Log[x])/x^(n*p*q)

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \log (x) \]

[In]

Integrate[x^(-1 - n*p*q)*(a*(b*x^n)^p)^q,x]

[Out]

((a*(b*x^n)^p)^q*Log[x])/x^(n*p*q)

Maple [F]

\[\int x^{-n p q -1} {\left (a \left (b \,x^{n}\right )^{p}\right )}^{q}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \log \left (x\right ) \]

[In]

integrate(x^(-n*p*q-1)*(a*(b*x^n)^p)^q,x, algorithm="fricas")

[Out]

e^(p*q*log(b) + q*log(a))*log(x)

Sympy [A] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=x x^{- n p q - 1} \left (a \left (b x^{n}\right )^{p}\right )^{q} \log {\left (x \right )} \]

[In]

integrate(x**(-n*p*q-1)*(a*(b*x**n)**p)**q,x)

[Out]

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

Maxima [F]

\[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\int { \left (\left (b x^{n}\right )^{p} a\right )^{q} x^{-n p q - 1} \,d x } \]

[In]

integrate(x^(-n*p*q-1)*(a*(b*x^n)^p)^q,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\int { \left (\left (b x^{n}\right )^{p} a\right )^{q} x^{-n p q - 1} \,d x } \]

[In]

integrate(x^(-n*p*q-1)*(a*(b*x^n)^p)^q,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^{-1-n p q} \left (a \left (b x^n\right )^p\right )^q \, dx=\int \frac {{\left (a\,{\left (b\,x^n\right )}^p\right )}^q}{x^{n\,p\,q+1}} \,d x \]

[In]

int((a*(b*x^n)^p)^q/x^(n*p*q + 1),x)

[Out]

int((a*(b*x^n)^p)^q/x^(n*p*q + 1), x)