Optimal. Leaf size=21 \[ \log (x) x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0436247, antiderivative size = 21, normalized size of antiderivative = 1., 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 = {6679, 29} \[ \log (x) x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6679
Rule 29
Rubi steps
\begin{align*} \int x^{-1-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} \, dx\\ &=x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \log (x)\\ \end{align*}
Mathematica [A] time = 0.0050226, size = 21, normalized size = 1. \[ \log (x) x^{-n p q} \left (a \left (b x^n\right )^p\right )^q \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.171, size = 0, normalized size = 0. \begin{align*} \int{x}^{-npq-1} \left ( a \left ( b{x}^{n} \right ) ^{p} \right ) ^{q}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (b x^{n}\right )^{p} a\right )^{q} x^{-n p q - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.87412, size = 46, normalized size = 2.19 \begin{align*} e^{\left (p q \log \left (b\right ) + q \log \left (a\right )\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{- n p q - 1} \left (a \left (b x^{n}\right )^{p}\right )^{q}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (b x^{n}\right )^{p} a\right )^{q} x^{-n p q - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]