3.6 \(\int F^{c (a+b x)} \, dx\)

Optimal. Leaf size=20 \[ \frac{F^{c (a+b x)}}{b c \log (F)} \]

[Out]

F^(c*(a + b*x))/(b*c*Log[F])

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Rubi [A]  time = 0.0036889, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2194} \[ \frac{F^{c (a+b x)}}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x)),x]

[Out]

F^(c*(a + b*x))/(b*c*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 F^{c (a+b x)} \, dx &=\frac{F^{c (a+b x)}}{b c \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0048603, size = 21, normalized size = 1.05 \[ \frac{F^{a c+b c x}}{b c \log (F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(c*(a + b*x)),x]

[Out]

F^(a*c + b*c*x)/(b*c*Log[F])

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Maple [A]  time = 0.001, size = 21, normalized size = 1.1 \begin{align*}{\frac{{F}^{c \left ( bx+a \right ) }}{bc\ln \left ( F \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(b*x+a)),x)

[Out]

F^(c*(b*x+a))/b/c/ln(F)

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Maxima [A]  time = 0.944983, size = 27, normalized size = 1.35 \begin{align*} \frac{F^{{\left (b x + a\right )} c}}{b c \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a)),x, algorithm="maxima")

[Out]

F^((b*x + a)*c)/(b*c*log(F))

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Fricas [A]  time = 1.51071, size = 41, normalized size = 2.05 \begin{align*} \frac{F^{b c x + a c}}{b c \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a)),x, algorithm="fricas")

[Out]

F^(b*c*x + a*c)/(b*c*log(F))

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Sympy [A]  time = 0.108263, size = 20, normalized size = 1. \begin{align*} \begin{cases} \frac{F^{c \left (a + b x\right )}}{b c \log{\left (F \right )}} & \text{for}\: b c \log{\left (F \right )} \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(c*(b*x+a)),x)

[Out]

Piecewise((F**(c*(a + b*x))/(b*c*log(F)), Ne(b*c*log(F), 0)), (x, True))

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Giac [A]  time = 3.58521, size = 27, normalized size = 1.35 \begin{align*} \frac{F^{{\left (b x + a\right )} c}}{b c \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a)),x, algorithm="giac")

[Out]

F^((b*x + a)*c)/(b*c*log(F))