Optimal. Leaf size=202 \[ -\frac{a^2}{d (c+d x)}+\frac{2 a b f g n \log (F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )}{d^2}-\frac{2 a b \left (F^{e g+f g x}\right )^n}{d (c+d x)}+\frac{2 b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right )}{d^2}-\frac{b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)} \]
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Rubi [A] time = 0.341421, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2183, 2177, 2182, 2178} \[ -\frac{a^2}{d (c+d x)}+\frac{2 a b f g n \log (F) \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )}{d^2}-\frac{2 a b \left (F^{e g+f g x}\right )^n}{d (c+d x)}+\frac{2 b^2 f g n \log (F) \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right )}{d^2}-\frac{b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)} \]
Antiderivative was successfully verified.
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Rule 2183
Rule 2177
Rule 2182
Rule 2178
Rubi steps
\begin{align*} \int \frac{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}{(c+d x)^2} \, dx &=\int \left (\frac{a^2}{(c+d x)^2}+\frac{2 a b \left (F^{e g+f g x}\right )^n}{(c+d x)^2}+\frac{b^2 \left (F^{e g+f g x}\right )^{2 n}}{(c+d x)^2}\right ) \, dx\\ &=-\frac{a^2}{d (c+d x)}+(2 a b) \int \frac{\left (F^{e g+f g x}\right )^n}{(c+d x)^2} \, dx+b^2 \int \frac{\left (F^{e g+f g x}\right )^{2 n}}{(c+d x)^2} \, dx\\ &=-\frac{a^2}{d (c+d x)}-\frac{2 a b \left (F^{e g+f g x}\right )^n}{d (c+d x)}-\frac{b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)}+\frac{(2 a b f g n \log (F)) \int \frac{\left (F^{e g+f g x}\right )^n}{c+d x} \, dx}{d}+\frac{\left (2 b^2 f g n \log (F)\right ) \int \frac{\left (F^{e g+f g x}\right )^{2 n}}{c+d x} \, dx}{d}\\ &=-\frac{a^2}{d (c+d x)}-\frac{2 a b \left (F^{e g+f g x}\right )^n}{d (c+d x)}-\frac{b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)}+\frac{\left (2 a b f F^{-n (e g+f g x)} \left (F^{e g+f g x}\right )^n g n \log (F)\right ) \int \frac{F^{n (e g+f g x)}}{c+d x} \, dx}{d}+\frac{\left (2 b^2 f F^{-2 n (e g+f g x)} \left (F^{e g+f g x}\right )^{2 n} g n \log (F)\right ) \int \frac{F^{2 n (e g+f g x)}}{c+d x} \, dx}{d}\\ &=-\frac{a^2}{d (c+d x)}-\frac{2 a b \left (F^{e g+f g x}\right )^n}{d (c+d x)}-\frac{b^2 \left (F^{e g+f g x}\right )^{2 n}}{d (c+d x)}+\frac{2 a b f F^{\left (e-\frac{c f}{d}\right ) g n-g n (e+f x)} \left (F^{e g+f g x}\right )^n g n \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right ) \log (F)}{d^2}+\frac{2 b^2 f 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} g n \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right ) \log (F)}{d^2}\\ \end{align*}
Mathematica [A] time = 0.683578, size = 136, normalized size = 0.67 \[ \frac{2 a b f g n \log (F) \left (F^{g (e+f x)}\right )^n F^{-\frac{f g n (c+d x)}{d}} \text{Ei}\left (\frac{f g n (c+d x) \log (F)}{d}\right )-\frac{d \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}{c+d x}+2 b^2 f g n \log (F) \left (F^{g (e+f x)}\right )^{2 n} F^{-\frac{2 f g n (c+d x)}{d}} \text{Ei}\left (\frac{2 f g n (c+d x) \log (F)}{d}\right )}{d^2} \]
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 ) ^{2}}{ \left ( dx+c \right ) ^{2}}}\, 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 )}^{2 \, n} b^{2} \int \frac{{\left (F^{f g x}\right )}^{2 \, n}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} + 2 \,{\left (F^{e g}\right )}^{n} a b \int \frac{{\left (F^{f g x}\right )}^{n}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} - \frac{a^{2}}{d^{2} x + c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5317, size = 389, normalized size = 1.93 \begin{align*} -\frac{2 \, F^{f g n x + e g n} a b d + F^{2 \, f g n x + 2 \, e g n} b^{2} d - 2 \,{\left (b^{2} d f g n x + b^{2} c f g n\right )} F^{\frac{2 \,{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{2 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right ) - 2 \,{\left (a b d f g n x + a b c f g n\right )} F^{\frac{{\left (d e - c f\right )} g n}{d}}{\rm Ei}\left (\frac{{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) \log \left (F\right ) + a^{2} d}{d^{3} x + c d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \left (F^{e g} F^{f g x}\right )^{n}\right )^{2}}{\left (c + d x\right )^{2}}\, dx \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 )}^{2}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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