Optimal. Leaf size=29 \[ \text {Int}\left (\frac {1}{(c+d x) \left (a+b \left (F^{e g+f g x}\right )^n\right )^2},x\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx &=\int \frac {1}{\left (a+b \left (F^{e g+f g x}\right )^n\right )^2 (c+d x)} \, dx\\ \end {align*}
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Mathematica [A] time = 1.01, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} d x + a^{2} c + {\left (b^{2} d x + b^{2} c\right )} {\left (F^{f g x + e g}\right )}^{2 \, n} + 2 \, {\left (a b d x + a b c\right )} {\left (F^{f g x + e g}\right )}^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right )^{2} \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{a^{2} d f g n x \log \relax (F) + a^{2} c f g n \log \relax (F) + {\left ({\left (F^{e g}\right )}^{n} a b d f g n x \log \relax (F) + {\left (F^{e g}\right )}^{n} a b c f g n \log \relax (F)\right )} {\left (F^{f g x}\right )}^{n}} + \int \frac {d f g n x \log \relax (F) + c f g n \log \relax (F) + d}{a^{2} d^{2} f g n x^{2} \log \relax (F) + 2 \, a^{2} c d f g n x \log \relax (F) + a^{2} c^{2} f g n \log \relax (F) + {\left ({\left (F^{e g}\right )}^{n} a b d^{2} f g n x^{2} \log \relax (F) + 2 \, {\left (F^{e g}\right )}^{n} a b c d f g n x \log \relax (F) + {\left (F^{e g}\right )}^{n} a b c^{2} f g n \log \relax (F)\right )} {\left (F^{f g x}\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^2\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{a^{2} c f g n \log {\relax (F )} + a^{2} d f g n x \log {\relax (F )} + \left (a b c f g n \log {\relax (F )} + a b d f g n x \log {\relax (F )}\right ) \left (F^{g \left (e + f x\right )}\right )^{n}} + \frac {\int \frac {d}{a c^{2} + 2 a c d x + a d^{2} x^{2} + b c^{2} e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}} + 2 b c d x e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}} + b d^{2} x^{2} e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\, dx + \int \frac {c f g n \log {\relax (F )}}{a c^{2} + 2 a c d x + a d^{2} x^{2} + b c^{2} e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}} + 2 b c d x e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}} + b d^{2} x^{2} e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\, dx + \int \frac {d f g n x \log {\relax (F )}}{a c^{2} + 2 a c d x + a d^{2} x^{2} + b c^{2} e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}} + 2 b c d x e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}} + b d^{2} x^{2} e^{e g n \log {\relax (F )}} e^{f g n x \log {\relax (F )}}}\, dx}{a f g n \log {\relax (F )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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