Optimal. Leaf size=41 \[ \frac{\left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1}}{b d e n (p+1) \log (F)} \]
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Rubi [A] time = 0.0646444, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2246, 32} \[ \frac{\left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1}}{b d e n (p+1) \log (F)} \]
Antiderivative was successfully verified.
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Rule 2246
Rule 32
Rubi steps
\begin{align*} \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int (a+b x)^p \, dx,x,\left (F^{e (c+d x)}\right )^n\right )}{d e n \log (F)}\\ &=\frac{\left (a+b \left (F^{e (c+d x)}\right )^n\right )^{1+p}}{b d e n (1+p) \log (F)}\\ \end{align*}
Mathematica [F] time = 0.289949, size = 0, normalized size = 0. \[ \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.008, size = 42, normalized size = 1. \begin{align*}{\frac{ \left ( a+b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n} \right ) ^{1+p}}{bden \left ( 1+p \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 = 1.56138, size = 122, normalized size = 2.98 \begin{align*} \frac{{\left (F^{d e n x + c e n} b + a\right )}{\left (F^{d e n x + c e n} b + a\right )}^{p}}{{\left (b d e n p + b d e n\right )} \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.35188, size = 58, normalized size = 1.41 \begin{align*} \frac{{\left (F^{d n x e + c n e} b + a\right )}^{p + 1} e^{\left (-1\right )}}{b d n{\left (p + 1\right )} \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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