Optimal. Leaf size=24 \[ \frac{\left (a+b F^x\right )^{n+1}}{b (n+1) \log (F)} \]
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Rubi [A] time = 0.0229501, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2246, 32} \[ \frac{\left (a+b F^x\right )^{n+1}}{b (n+1) \log (F)} \]
Antiderivative was successfully verified.
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Rule 2246
Rule 32
Rubi steps
\begin{align*} \int F^x \left (a+b F^x\right )^n \, dx &=\frac{\operatorname{Subst}\left (\int (a+b x)^n \, dx,x,F^x\right )}{\log (F)}\\ &=\frac{\left (a+b F^x\right )^{1+n}}{b (1+n) \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0240604, size = 24, normalized size = 1. \[ \frac{\left (a+b F^x\right )^{n+1}}{b n \log (F)+b \log (F)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 1. \begin{align*}{\frac{ \left ( a+b{F}^{x} \right ) ^{1+n}}{b \left ( 1+n \right ) \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 = 1.55113, size = 62, normalized size = 2.58 \begin{align*} \frac{{\left (F^{x} b + a\right )}{\left (F^{x} b + a\right )}^{n}}{{\left (b n + b\right )} \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.425, size = 82, normalized size = 3.42 \begin{align*} \begin{cases} \frac{x}{a} & \text{for}\: F = 1 \wedge b = 0 \wedge n = -1 \\x \left (a + b\right )^{n} & \text{for}\: F = 1 \\\frac{F^{x} a^{n}}{\log{\left (F \right )}} & \text{for}\: b = 0 \\\frac{\log{\left (F^{x} + \frac{a}{b} \right )}}{b \log{\left (F \right )}} & \text{for}\: n = -1 \\\frac{F^{x} b \left (F^{x} b + a\right )^{n}}{b n \log{\left (F \right )} + b \log{\left (F \right )}} + \frac{a \left (F^{x} b + a\right )^{n}}{b n \log{\left (F \right )} + b \log{\left (F \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26729, size = 32, normalized size = 1.33 \begin{align*} \frac{{\left (F^{x} b + a\right )}^{n + 1}}{b{\left (n + 1\right )} \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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