Optimal. Leaf size=228 \[ \frac {a^2 (c+d x)^{1+m}}{d (1+m)}+\frac {2^{-1-m} b^2 F^{2 \left (e-\frac {c f}{d}\right ) g n-2 g n (e+f x)} \left (F^{e g+f g x}\right )^{2 n} (c+d x)^m \Gamma \left (1+m,-\frac {2 f g n (c+d x) \log (F)}{d}\right ) \left (-\frac {f g n (c+d x) \log (F)}{d}\right )^{-m}}{f g n \log (F)}+\frac {2 a b F^{\left (e-\frac {c f}{d}\right ) g n-g n (e+f x)} \left (F^{e g+f g x}\right )^n (c+d x)^m \Gamma \left (1+m,-\frac {f g n (c+d x) \log (F)}{d}\right ) \left (-\frac {f g n (c+d x) \log (F)}{d}\right )^{-m}}{f g n \log (F)} \]
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Rubi [A]
time = 0.21, 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 = {2214, 2213,
2212} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 2213
Rule 2214
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 = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )^{2} \left (d x +c \right )^{m}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.11, size = 197, normalized size = 0.86 \begin {gather*} \frac {4 \, {\left (a b d m + a b d\right )} e^{\left (-\frac {d m \log \left (-\frac {f g n \log \left (F\right )}{d}\right ) + {\left (c f g n - d g n e\right )} \log \left (F\right )}{d}\right )} \Gamma \left (m + 1, -\frac {{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + {\left (b^{2} d m + b^{2} d\right )} e^{\left (-\frac {d m \log \left (-\frac {2 \, f g n \log \left (F\right )}{d}\right ) + 2 \, {\left (c f g n - d g n e\right )} \log \left (F\right )}{d}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (d f g n x + c f g n\right )} \log \left (F\right )}{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 \left (F\right )}{2 \, {\left (d f g m + d f g\right )} n \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^2\,{\left (c+d\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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