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

Optimal. Leaf size=15 \[ \frac{F^{a+b x}}{b \log (F)} \]

[Out]

F^(a + b*x)/(b*Log[F])

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Rubi [A]  time = 0.0028422, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2194} \[ \frac{F^{a+b x}}{b \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*x),x]

[Out]

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

Mathematica [A]  time = 0.0035217, size = 15, normalized size = 1. \[ \frac{F^{a+b x}}{b \log (F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*x),x]

[Out]

F^(a + b*x)/(b*Log[F])

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Maple [A]  time = 0.006, size = 16, normalized size = 1.1 \begin{align*}{\frac{{F}^{bx+a}}{b\ln \left ( F \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05093, size = 20, normalized size = 1.33 \begin{align*} \frac{F^{b x + a}}{b \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

F^(b*x + a)/(b*log(F))

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Fricas [A]  time = 1.51011, size = 32, normalized size = 2.13 \begin{align*} \frac{F^{b x + a}}{b \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

F^(b*x + a)/(b*log(F))

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Sympy [A]  time = 0.092784, size = 15, normalized size = 1. \begin{align*} \begin{cases} \frac{F^{a + b x}}{b \log{\left (F \right )}} & \text{for}\: b \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**(b*x+a),x)

[Out]

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

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Giac [A]  time = 1.19125, size = 20, normalized size = 1.33 \begin{align*} \frac{F^{b x + a}}{b \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

F^(b*x + a)/(b*log(F))