Optimal. Leaf size=228 \[ \frac {a^2 (c+d x)^{m+1}}{d (m+1)}+\frac {2 a b (c+d x)^m \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac {c f}{d}\right )-g n (e+f x)} \left (-\frac {f g n \log (F) (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f g n (c+d x) \log (F)}{d}\right )}{f g n \log (F)}+\frac {b^2 2^{-m-1} (c+d x)^m \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac {c f}{d}\right )-2 g n (e+f x)} \left (-\frac {f g n \log (F) (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 f g n (c+d x) \log (F)}{d}\right )}{f g n \log (F)} \]
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Rubi [A] time = 0.28, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2183, 2182, 2181} \[ \frac {2 a b (c+d x)^m \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac {c f}{d}\right )-g n (e+f x)} \left (-\frac {f g n \log (F) (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {f g n \log (F) (c+d x)}{d}\right )}{f g n \log (F)}+\frac {b^2 2^{-m-1} (c+d x)^m \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac {c f}{d}\right )-2 g n (e+f x)} \left (-\frac {f g n \log (F) (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {2 f g n \log (F) (c+d x)}{d}\right )}{f g n \log (F)}+\frac {a^2 (c+d x)^{m+1}}{d (m+1)} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 2182
Rule 2183
Rubi steps
\begin {align*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx &=\int \left (a^2 (c+d x)^m+2 a b \left (F^{e g+f g x}\right )^n (c+d x)^m+b^2 \left (F^{e g+f g x}\right )^{2 n} (c+d x)^m\right ) \, dx\\ &=\frac {a^2 (c+d x)^{1+m}}{d (1+m)}+(2 a b) \int \left (F^{e g+f g x}\right )^n (c+d x)^m \, dx+b^2 \int \left (F^{e g+f g x}\right )^{2 n} (c+d x)^m \, dx\\ &=\frac {a^2 (c+d x)^{1+m}}{d (1+m)}+\left (2 a b F^{-n (e g+f g x)} \left (F^{e g+f g x}\right )^n\right ) \int F^{n (e g+f g x)} (c+d x)^m \, dx+\left (b^2 F^{-2 n (e g+f g x)} \left (F^{e g+f g x}\right )^{2 n}\right ) \int F^{2 n (e g+f g x)} (c+d x)^m \, dx\\ &=\frac {a^2 (c+d x)^{1+m}}{d (1+m)}+\frac {2^{-1-m} 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} (c+d x)^m \Gamma \left (1+m,-\frac {2 f g n (c+d x) \log (F)}{d}\right ) \left (-\frac {f g n (c+d x) \log (F)}{d}\right )^{-m}}{f g n \log (F)}+\frac {2 a 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 (c+d x)^m \Gamma \left (1+m,-\frac {f g n (c+d x) \log (F)}{d}\right ) \left (-\frac {f g n (c+d x) \log (F)}{d}\right )^{-m}}{f g n \log (F)}\\ \end {align*}
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Mathematica [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.45, size = 191, normalized size = 0.84 \[ \frac {4 \, {\left (a b d m + a b d\right )} e^{\left (\frac {{\left (d e - c f\right )} g n \log \relax (F) - d m \log \left (-\frac {f g n \log \relax (F)}{d}\right )}{d}\right )} \Gamma \left (m + 1, -\frac {{\left (d f g n x + c f g n\right )} \log \relax (F)}{d}\right ) + {\left (b^{2} d m + b^{2} d\right )} e^{\left (\frac {2 \, {\left (d e - c f\right )} g n \log \relax (F) - d m \log \left (-\frac {2 \, f g n \log \relax (F)}{d}\right )}{d}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (d f g n x + c f g n\right )} \log \relax (F)}{d}\right ) + 2 \, {\left (a^{2} d f g n x + a^{2} c f g n\right )} {\left (d x + c\right )}^{m} \log \relax (F)}{2 \, {\left (d f g m + d f g\right )} n \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2} {\left (d x + c\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (b \left (F^{\left (f x +e \right ) g}\right )^{n}+a \right )^{2} \left (d x +c \right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (F^{e g}\right )}^{2 \, n} b^{2} \int e^{\left (m \log \left (d x + c\right ) + 2 \, n \log \left (F^{f g x}\right )\right )}\,{d x} + 2 \, {\left (F^{e g}\right )}^{n} a b \int e^{\left (m \log \left (d x + c\right ) + n \log \left (F^{f g x}\right )\right )}\,{d x} + \frac {{\left (d x + c\right )}^{m + 1} a^{2}}{d {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^2\,{\left (c+d\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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