Optimal. Leaf size=37 \[ \frac{e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{p+1}}{b n (p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0589087, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2247, 2246, 32} \[ \frac{e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{p+1}}{b n (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2247
Rule 2246
Rule 32
Rubi steps
\begin{align*} \int e^{n x} \left (a+b \left (e^x\right )^n\right )^p \, dx &=\left (e^{n x} \left (e^x\right )^{-n}\right ) \int \left (e^x\right )^n \left (a+b \left (e^x\right )^n\right )^p \, dx\\ &=\frac{\left (e^{n x} \left (e^x\right )^{-n}\right ) \operatorname{Subst}\left (\int (a+b x)^p \, dx,x,\left (e^x\right )^n\right )}{n}\\ &=\frac{e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{1+p}}{b n (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0424461, size = 36, normalized size = 0.97 \[ \frac{e^{n x} \left (e^x\right )^{-n} \left (a+b \left (e^x\right )^n\right )^{p+1}}{b n p+b n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.015, size = 52, normalized size = 1.4 \begin{align*}{\frac{{{\rm e}^{nx}}{{\rm e}^{p\ln \left ( a+b{{\rm e}^{nx}} \right ) }}}{n \left ( 1+p \right ) }}+{\frac{a{{\rm e}^{p\ln \left ( a+b{{\rm e}^{nx}} \right ) }}}{bn \left ( 1+p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.57852, size = 66, normalized size = 1.78 \begin{align*} \frac{{\left (b e^{\left (n x\right )} + a\right )}{\left (b e^{\left (n x\right )} + a\right )}^{p}}{b n p + b n} \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 \left (a + b \left (e^{x}\right )^{n}\right )^{p} e^{n x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.26427, size = 32, normalized size = 0.86 \begin{align*} \frac{{\left (b e^{\left (n x\right )} + a\right )}^{p + 1}}{b n{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]