Optimal. Leaf size=37 \[ -\frac{(c x)^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0123034, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {264} \[ -\frac{(c x)^{-n (p+1)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rubi steps
\begin{align*} \int (c x)^{-1-n-n p} \left (a+b x^n\right )^p \, dx &=-\frac{(c x)^{-n (1+p)} \left (a+b x^n\right )^{1+p}}{a c n (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0253412, size = 37, normalized size = 1. \[ -\frac{x (c x)^{-n (p+1)-1} \left (a+b x^n\right )^{p+1}}{a n (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.076, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{-np-n-1} \left ( a+b{x}^{n} \right ) ^{p}\, 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 (b x^{n} + a\right )}^{p} \left (c x\right )^{-n p - n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.42492, size = 194, normalized size = 5.24 \begin{align*} -\frac{{\left (b x x^{n} e^{\left (-{\left (n p + n + 1\right )} \log \left (c\right ) -{\left (n p + n + 1\right )} \log \left (x\right )\right )} + a x e^{\left (-{\left (n p + n + 1\right )} \log \left (c\right ) -{\left (n p + n + 1\right )} \log \left (x\right )\right )}\right )}{\left (b x^{n} + a\right )}^{p}}{a n p + a n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (b x^{n} + a\right )}^{p} \left (c x\right )^{-n p - n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]