Optimal. Leaf size=200 \[ \frac{3 a^2 b \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )}{d}+\frac{a^3 \log (c+d x)}{d}+\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right )}{d}+\frac{b^3 \left (F^{e g+f g x}\right )^{3 n} F^{3 g n \left (e-\frac{c f}{d}\right )-3 g n (e+f x)} \text{Ei}\left (\frac{3 f g n (c+d x) \log (F)}{d}\right )}{d} \]
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Rubi [A] time = 0.3538, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2183, 2182, 2178} \[ \frac{3 a^2 b \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )}{d}+\frac{a^3 \log (c+d x)}{d}+\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right )}{d}+\frac{b^3 \left (F^{e g+f g x}\right )^{3 n} F^{3 g n \left (e-\frac{c f}{d}\right )-3 g n (e+f x)} \text{Ei}\left (\frac{3 f g n (c+d x) \log (F)}{d}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2183
Rule 2182
Rule 2178
Rubi steps
\begin{align*} \int \frac{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3}{c+d x} \, dx &=\int \left (\frac{a^3}{c+d x}+\frac{3 a^2 b \left (F^{e g+f g x}\right )^n}{c+d x}+\frac{3 a b^2 \left (F^{e g+f g x}\right )^{2 n}}{c+d x}+\frac{b^3 \left (F^{e g+f g x}\right )^{3 n}}{c+d x}\right ) \, dx\\ &=\frac{a^3 \log (c+d x)}{d}+\left (3 a^2 b\right ) \int \frac{\left (F^{e g+f g x}\right )^n}{c+d x} \, dx+\left (3 a b^2\right ) \int \frac{\left (F^{e g+f g x}\right )^{2 n}}{c+d x} \, dx+b^3 \int \frac{\left (F^{e g+f g x}\right )^{3 n}}{c+d x} \, dx\\ &=\frac{a^3 \log (c+d x)}{d}+\left (3 a^2 b F^{-n (e g+f g x)} \left (F^{e g+f g x}\right )^n\right ) \int \frac{F^{n (e g+f g x)}}{c+d x} \, dx+\left (3 a b^2 F^{-2 n (e g+f g x)} \left (F^{e g+f g x}\right )^{2 n}\right ) \int \frac{F^{2 n (e g+f g x)}}{c+d x} \, dx+\left (b^3 F^{-3 n (e g+f g x)} \left (F^{e g+f g x}\right )^{3 n}\right ) \int \frac{F^{3 n (e g+f g x)}}{c+d x} \, dx\\ &=\frac{3 a^2 b F^{\left (e-\frac{c f}{d}\right ) g n-g n (e+f x)} \left (F^{e g+f g x}\right )^n \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )}{d}+\frac{3 a b^2 F^{2 \left (e-\frac{c f}{d}\right ) g n-2 g n (e+f x)} \left (F^{e g+f g x}\right )^{2 n} \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right )}{d}+\frac{b^3 F^{3 \left (e-\frac{c f}{d}\right ) g n-3 g n (e+f x)} \left (F^{e g+f g x}\right )^{3 n} \text{Ei}\left (\frac{3 f g n (c+d x) \log (F)}{d}\right )}{d}+\frac{a^3 \log (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.477363, size = 160, normalized size = 0.8 \[ \frac{3 a^2 b \left (F^{g (e+f x)}\right )^n F^{-\frac{f g n (c+d x)}{d}} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )+a^3 \log (c+d x)+3 a b^2 \left (F^{g (e+f x)}\right )^{2 n} F^{-\frac{2 f g n (c+d x)}{d}} \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right )+b^3 \left (F^{g (e+f x)}\right )^{3 n} F^{-\frac{3 f g n (c+d x)}{d}} \text{Ei}\left (\frac{3 f g n (c+d x) \log (F)}{d}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{3}}{dx+c}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\left (F^{e g}\right )}^{3 \, n} b^{3} \int \frac{{\left (F^{f g x}\right )}^{3 \, n}}{d x + c}\,{d x} + 3 \,{\left (F^{e g}\right )}^{2 \, n} a b^{2} \int \frac{{\left (F^{f g x}\right )}^{2 \, n}}{d x + c}\,{d x} + 3 \,{\left (F^{e g}\right )}^{n} a^{2} b \int \frac{{\left (F^{f g x}\right )}^{n}}{d x + c}\,{d x} + \frac{a^{3} \log \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54426, size = 306, normalized size = 1.53 \begin{align*} \frac{F^{\frac{3 \,{\left (d e - c f\right )} g n}{d}} b^{3}{\rm Ei}\left (\frac{3 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + 3 \, F^{\frac{2 \,{\left (d e - c f\right )} g n}{d}} a b^{2}{\rm Ei}\left (\frac{2 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + 3 \, F^{\frac{{\left (d e - c f\right )} g n}{d}} a^{2} b{\rm Ei}\left (\frac{{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + a^{3} \log \left (d x + c\right )}{d} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{3}}{d x + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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