Optimal. Leaf size=77 \[ \frac{a (c+d x)^2}{2 d}+\frac{b (c+d x) \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.135804, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{a (c+d x)^2}{2 d}+\frac{b (c+d x) \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac{b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*(F^(g*(e + f*x)))^n)*(c + d*x),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.937, size = 65, normalized size = 0.84 \[ \frac{a \left (c + d x\right )^{2}}{2 d} - \frac{b d \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}} + \frac{b \left (c + d x\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{f g n \log{\left (F \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*(F**(g*(f*x+e)))**n)*(d*x+c),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.168407, size = 73, normalized size = 0.95 \[ \frac{1}{2} a x (2 c+d x)+\frac{b (c+d x) \left (F^{g (e+f x)}\right )^n}{f g n \log (F)}-\frac{b d \left (F^{g (e+f x)}\right )^n}{f^2 g^2 n^2 \log ^2(F)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*(F^(g*(e + f*x)))^n)*(c + d*x),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.03, size = 105, normalized size = 1.4 \[ acx+{\frac{b{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}c}{ngf\ln \left ( F \right ) }}-{\frac{b{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}d}{ \left ( \ln \left ( F \right ) \right ) ^{2}{f}^{2}{g}^{2}{n}^{2}}}+{\frac{bdx{{\rm e}^{n\ln \left ({{\rm e}^{g \left ( fx+e \right ) \ln \left ( F \right ) }} \right ) }}}{ngf\ln \left ( F \right ) }}+{\frac{ad{x}^{2}}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*(F^(g*(f*x+e)))^n)*(d*x+c),x)
[Out]
_______________________________________________________________________________________
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),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.279655, size = 117, normalized size = 1.52 \[ \frac{{\left (a d f^{2} g^{2} n^{2} x^{2} + 2 \, a c f^{2} g^{2} n^{2} x\right )} \log \left (F\right )^{2} - 2 \,{\left (b d -{\left (b d f g n x + b c f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{2 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((F^((f*x + e)*g))^n*b + a)*(d*x + c),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.397601, size = 94, normalized size = 1.22 \[ a c x + \frac{a d x^{2}}{2} + \begin{cases} \frac{\left (b c f g n \log{\left (F \right )} + b d f g n x \log{\left (F \right )} - b d\right ) \left (F^{g \left (e + f x\right )}\right )^{n}}{f^{2} g^{2} n^{2} \log{\left (F \right )}^{2}} & \text{for}\: f^{2} g^{2} n^{2} \log{\left (F \right )}^{2} \neq 0 \\b c x + \frac{b d x^{2}}{2} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*(F**(g*(f*x+e)))**n)*(d*x+c),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.275421, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((F^((f*x + e)*g))^n*b + a)*(d*x + c),x, algorithm="giac")
[Out]