Optimal. Leaf size=31 \[ \frac{\left (a+b F^{c+d x}\right )^{n+1}}{b d (n+1) \log (F)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0351259, antiderivative size = 31, 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 = {2246, 32} \[ \frac{\left (a+b F^{c+d x}\right )^{n+1}}{b d (n+1) \log (F)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2246
Rule 32
Rubi steps
\begin{align*} \int F^{c+d x} \left (a+b F^{c+d x}\right )^n \, dx &=\frac{\operatorname{Subst}\left (\int (a+b x)^n \, dx,x,F^{c+d x}\right )}{d \log (F)}\\ &=\frac{\left (a+b F^{c+d x}\right )^{1+n}}{b d (1+n) \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0303065, size = 30, normalized size = 0.97 \[ \frac{\left (a+b F^{c+d x}\right )^{n+1}}{b d n \log (F)+b d \log (F)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 32, normalized size = 1. \begin{align*}{\frac{ \left ( a+b{F}^{dx+c} \right ) ^{1+n}}{bd \left ( 1+n \right ) \ln \left ( F \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 = 1.53902, size = 89, normalized size = 2.87 \begin{align*} \frac{{\left (F^{d x + c} b + a\right )}{\left (F^{d x + c} b + a\right )}^{n}}{{\left (b d n + b d\right )} \log \left (F\right )} \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 [A] time = 1.22773, size = 42, normalized size = 1.35 \begin{align*} \frac{{\left (F^{d x + c} b + a\right )}^{n + 1}}{b d{\left (n + 1\right )} \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]