Optimal. Leaf size=30 \[ a x+\frac{b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)} \]
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Rubi [A] time = 0.0158124, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2194} \[ a x+\frac{b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)} \]
Antiderivative was successfully verified.
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Rule 2194
Rubi steps
\begin{align*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) \, dx &=a x+b \int \left (F^{g (e+f x)}\right )^n \, dx\\ &=a x+\frac{b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0254274, size = 30, normalized size = 1. \[ a x+\frac{b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 31, normalized size = 1. \begin{align*} ax+{\frac{b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n}}{ngf\ln \left ( F \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 1.64144, size = 80, normalized size = 2.67 \begin{align*} \frac{a f g n x \log \left (F\right ) + F^{f g n x + e g n} b}{f g n \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.267345, size = 32, normalized size = 1.07 \begin{align*} a x + \begin{cases} \frac{b \left (F^{g \left (e + f x\right )}\right )^{n}}{f g n \log{\left (F \right )}} & \text{for}\: f g n \log{\left (F \right )} \neq 0 \\b x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24996, size = 43, normalized size = 1.43 \begin{align*} a x + \frac{F^{f g n x + g n e} b}{f g n \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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