3.27 \(\int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x) \, dx\)

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]

(a*(c + d*x)^2)/(2*d) - (b*d*(F^(e*g + f*g*x))^n)/(f^2*g^2*n^2*Log[F]^2) + (b*(F
^(e*g + f*g*x))^n*(c + d*x))/(f*g*n*Log[F])

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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]

(a*(c + d*x)^2)/(2*d) - (b*d*(F^(e*g + f*g*x))^n)/(f^2*g^2*n^2*Log[F]^2) + (b*(F
^(e*g + f*g*x))^n*(c + d*x))/(f*g*n*Log[F])

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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]

a*(c + d*x)**2/(2*d) - b*d*(F**(g*(e + f*x)))**n/(f**2*g**2*n**2*log(F)**2) + b*
(c + d*x)*(F**(g*(e + f*x)))**n/(f*g*n*log(F))

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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]

(a*x*(2*c + d*x))/2 - (b*d*(F^(g*(e + f*x)))^n)/(f^2*g^2*n^2*Log[F]^2) + (b*(F^(
g*(e + f*x)))^n*(c + d*x))/(f*g*n*Log[F])

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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]

a*c*x+b/n/g/f/ln(F)*exp(n*ln(exp(g*(f*x+e)*ln(F))))*c-b/n^2/g^2/f^2/ln(F)^2*exp(
n*ln(exp(g*(f*x+e)*ln(F))))*d+1/n/g/f/ln(F)*b*d*x*exp(n*ln(exp(g*(f*x+e)*ln(F)))
)+1/2*a*d*x^2

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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]

1/2*((a*d*f^2*g^2*n^2*x^2 + 2*a*c*f^2*g^2*n^2*x)*log(F)^2 - 2*(b*d - (b*d*f*g*n*
x + b*c*f*g*n)*log(F))*F^(f*g*n*x + e*g*n))/(f^2*g^2*n^2*log(F)^2)

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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]

a*c*x + a*d*x**2/2 + Piecewise(((b*c*f*g*n*log(F) + b*d*f*g*n*x*log(F) - b*d)*(F
**(g*(e + f*x)))**n/(f**2*g**2*n**2*log(F)**2), Ne(f**2*g**2*n**2*log(F)**2, 0))
, (b*c*x + b*d*x**2/2, True))

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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]

Done