3.189 \(\int x^m \left (a \left (b x^n\right )^p\right )^q \, dx\)

Optimal. Leaf size=26 \[ \frac{x^{m+1} \left (a \left (b x^n\right )^p\right )^q}{m+n p q+1} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.080876, antiderivative size = 26, 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 \[ \frac{x^{m+1} \left (a \left (b x^n\right )^p\right )^q}{m+n p q+1} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a*(b*x^n)^p)^q,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.05711, size = 34, normalized size = 1.31 \[ \frac{x^{- n p q} x^{m + n p q + 1} \left (a \left (b x^{n}\right )^{p}\right )^{q}}{m + n p q + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(a*(b*x**n)**p)**q,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0172672, size = 26, normalized size = 1. \[ \frac{x^{m+1} \left (a \left (b x^n\right )^p\right )^q}{m+n p q+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a*(b*x^n)^p)^q,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 27, normalized size = 1. \[{\frac{{x}^{1+m} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}}{npq+m+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(a*(b*x^n)^p)^q,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.64882, size = 45, normalized size = 1.73 \[ \frac{a^{q}{\left (b^{p}\right )}^{q} x e^{\left (m \log \left (x\right ) + q \log \left ({\left (x^{n}\right )}^{p}\right )\right )}}{n p q + m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n)^p*a)^q*x^m,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.235759, size = 42, normalized size = 1.62 \[ \frac{x x^{m} e^{\left (n p q \log \left (x\right ) + p q \log \left (b\right ) + q \log \left (a\right )\right )}}{n p q + m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n)^p*a)^q*x^m,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(a*(b*x**n)**p)**q,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224601, size = 43, normalized size = 1.65 \[ \frac{x e^{\left (n p q{\rm ln}\left (x\right ) + p q{\rm ln}\left (b\right ) + q{\rm ln}\left (a\right ) + m{\rm ln}\left (x\right )\right )}}{n p q + m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x^n)^p*a)^q*x^m,x, algorithm="giac")

[Out]

x*e^(n*p*q*ln(x) + p*q*ln(b) + q*ln(a) + m*ln(x))/(n*p*q + m + 1)