Optimal. Leaf size=116 \[ \frac{b (c+d x)^m \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \left (-\frac{f g n \log (F) (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f g n \log (F) (c+d x)}{d}\right )}{f g n \log (F)}+\frac{a (c+d x)^{m+1}}{d (m+1)} \]
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Rubi [A] time = 0.239244, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{b (c+d x)^m \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \left (-\frac{f g n \log (F) (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f g n \log (F) (c+d x)}{d}\right )}{f g n \log (F)}+\frac{a (c+d x)^{m+1}}{d (m+1)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*(F^(g*(e + f*x)))^n)*(c + d*x)^m,x]
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Rubi in Sympy [A] time = 22.9806, size = 105, normalized size = 0.91 \[ \frac{F^{g n \left (- e - f x\right )} F^{- \frac{g n \left (c f - d e\right )}{d}} b \left (\frac{f g n \left (- c - d x\right ) \log{\left (F \right )}}{d}\right )^{- m} \left (c + d x\right )^{m} \left (F^{g \left (e + f x\right )}\right )^{n} \Gamma{\left (m + 1,\frac{f g n \left (- c - d x\right ) \log{\left (F \right )}}{d} \right )}}{f g n \log{\left (F \right )}} + \frac{a \left (c + d x\right )^{m + 1}}{d \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(F**(g*(f*x+e)))**n)*(d*x+c)**m,x)
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Mathematica [A] time = 0.145095, size = 0, normalized size = 0. \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x)^m \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(a + b*(F^(g*(e + f*x)))^n)*(c + d*x)^m,x]
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Maple [F] time = 0.052, size = 0, normalized size = 0. \[ \int \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) \left ( dx+c \right ) ^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(F^(g*(f*x+e)))^n)*(d*x+c)^m,x)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((F^((f*x + e)*g))^n*b + a)*(d*x + c)^m,x, algorithm="maxima")
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Fricas [A] time = 0.271062, size = 150, normalized size = 1.29 \[ \frac{{\left (b d m + b d\right )} e^{\left (\frac{{\left (d e - c f\right )} g n \log \left (F\right ) - d m \log \left (-\frac{f g n \log \left (F\right )}{d}\right )}{d}\right )} \Gamma \left (m + 1, -\frac{{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) +{\left (a d f g n x + a c f g n\right )}{\left (d x + c\right )}^{m} \log \left (F\right )}{{\left (d f g m + d f g\right )} n \log \left (F\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((F^((f*x + e)*g))^n*b + a)*(d*x + c)^m,x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(F**(g*(f*x+e)))**n)*(d*x+c)**m,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}{\left (d x + c\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((F^((f*x + e)*g))^n*b + a)*(d*x + c)^m,x, algorithm="giac")
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