3.28 \(\int (a+b (F^{g (e+f x)})^n) \, dx\)

Optimal. Leaf size=30 \[ a x+\frac{b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)} \]

[Out]

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

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Rubi [A]  time = 0.0158124, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2194} \[ a x+\frac{b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[a + b*(F^(g*(e + f*x)))^n,x]

[Out]

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

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) \, dx &=a x+b \int \left (F^{g (e+f x)}\right )^n \, dx\\ &=a x+\frac{b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0254274, size = 30, normalized size = 1. \[ a x+\frac{b \left (F^{g (e+f x)}\right )^n}{f g n \log (F)} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*(F^(g*(e + f*x)))^n,x]

[Out]

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

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Maple [A]  time = 0.001, size = 31, normalized size = 1. \begin{align*} ax+{\frac{b \left ({F}^{g \left ( fx+e \right ) } \right ) ^{n}}{ngf\ln \left ( F \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*(F^(g*(f*x+e)))^n,x)

[Out]

a*x+b*(F^(g*(f*x+e)))^n/f/g/n/ln(F)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*(F^(g*(f*x+e)))^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.64144, size = 80, normalized size = 2.67 \begin{align*} \frac{a f g n x \log \left (F\right ) + F^{f g n x + e g n} b}{f g n \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*(F^(g*(f*x+e)))^n,x, algorithm="fricas")

[Out]

(a*f*g*n*x*log(F) + F^(f*g*n*x + e*g*n)*b)/(f*g*n*log(F))

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Sympy [A]  time = 0.267345, size = 32, normalized size = 1.07 \begin{align*} a x + \begin{cases} \frac{b \left (F^{g \left (e + f x\right )}\right )^{n}}{f g n \log{\left (F \right )}} & \text{for}\: f g n \log{\left (F \right )} \neq 0 \\b x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*(F**(g*(f*x+e)))**n,x)

[Out]

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

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Giac [A]  time = 1.24996, size = 43, normalized size = 1.43 \begin{align*} a x + \frac{F^{f g n x + g n e} b}{f g n \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*(F^(g*(f*x+e)))^n,x, algorithm="giac")

[Out]

a*x + F^(f*g*n*x + g*n*e)*b/(f*g*n*log(F))