Optimal. Leaf size=228 \[ \frac{2 a b (c+d x)^m \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \left (-\frac{f g n \log (F) (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f g n \log (F) (c+d x)}{d}\right )}{f g n \log (F)}+\frac{b^2 2^{-m-1} (c+d x)^m \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \left (-\frac{f g n \log (F) (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f g n \log (F) (c+d x)}{d}\right )}{f g n \log (F)}+\frac{a^2 (c+d x)^{m+1}}{d (m+1)} \]
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Rubi [A] time = 0.498875, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{2 a b (c+d x)^m \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac{c f}{d}\right )-g n (e+f x)} \left (-\frac{f g n \log (F) (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f g n \log (F) (c+d x)}{d}\right )}{f g n \log (F)}+\frac{b^2 2^{-m-1} (c+d x)^m \left (F^{e g+f g x}\right )^{2 n} F^{2 g n \left (e-\frac{c f}{d}\right )-2 g n (e+f x)} \left (-\frac{f g n \log (F) (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f g n \log (F) (c+d x)}{d}\right )}{f g n \log (F)}+\frac{a^2 (c+d x)^{m+1}}{d (m+1)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^m,x]
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Rubi in Sympy [A] time = 50.2532, size = 219, normalized size = 0.96 \[ \frac{F^{g n \left (- 2 e - 2 f x\right )} F^{- \frac{2 g n \left (c f - d e\right )}{d}} b^{2} \left (\frac{f g n \left (- 2 c - 2 d x\right ) \log{\left (F \right )}}{d}\right )^{- m} \left (c + d x\right )^{m} \left (F^{g \left (e + f x\right )}\right )^{2 n} \Gamma{\left (m + 1,\frac{f g n \left (- 2 c - 2 d x\right ) \log{\left (F \right )}}{d} \right )}}{2 f g n \log{\left (F \right )}} + \frac{2 F^{g n \left (- e - f x\right )} F^{- \frac{g n \left (c f - d e\right )}{d}} a b \left (\frac{f g n \left (- c - d x\right ) \log{\left (F \right )}}{d}\right )^{- m} \left (c + d x\right )^{m} \left (F^{g \left (e + f x\right )}\right )^{n} \Gamma{\left (m + 1,\frac{f g n \left (- c - d x\right ) \log{\left (F \right )}}{d} \right )}}{f g n \log{\left (F \right )}} + \frac{a^{2} \left (c + d x\right )^{m + 1}}{d \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(F**(g*(f*x+e)))**n)**2*(d*x+c)**m,x)
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Mathematica [A] time = 0.200101, size = 0, normalized size = 0. \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 (c+d x)^m \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(a + b*(F^(g*(e + f*x)))^n)^2*(c + d*x)^m,x]
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Maple [F] time = 0.051, size = 0, normalized size = 0. \[ \int \left ( a+b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n} \right ) ^{2} \left ( dx+c \right ) ^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(F^(g*(f*x+e)))^n)^2*(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)^2*(d*x + c)^m,x, algorithm="maxima")
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Fricas [A] time = 0.275973, size = 258, normalized size = 1.13 \[ \frac{4 \,{\left (a b d m + a b d\right )} e^{\left (\frac{{\left (d e - c f\right )} g n \log \left (F\right ) - d m \log \left (-\frac{f g n \log \left (F\right )}{d}\right )}{d}\right )} \Gamma \left (m + 1, -\frac{{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) +{\left (b^{2} d m + b^{2} d\right )} e^{\left (\frac{2 \,{\left (d e - c f\right )} g n \log \left (F\right ) - d m \log \left (-\frac{2 \, f g n \log \left (F\right )}{d}\right )}{d}\right )} \Gamma \left (m + 1, -\frac{2 \,{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + 2 \,{\left (a^{2} d f g n x + a^{2} c f g n\right )}{\left (d x + c\right )}^{m} \log \left (F\right )}{2 \,{\left (d f g m + d f g\right )} n \log \left (F\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((F^((f*x + e)*g))^n*b + a)^2*(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)**2*(d*x+c)**m,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2}{\left (d x + c\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((F^((f*x + e)*g))^n*b + a)^2*(d*x + c)^m,x, algorithm="giac")
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