3.248 \(\int b^x \, dx\)

Optimal. Leaf size=8 \[ \frac{b^x}{\log (b)} \]

[Out]

b^x/Log[b]

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Rubi [A]  time = 0.0020969, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 3, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2194} \[ \frac{b^x}{\log (b)} \]

Antiderivative was successfully verified.

[In]

Int[b^x,x]

[Out]

b^x/Log[b]

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

Mathematica [A]  time = 0.0005588, size = 8, normalized size = 1. \[ \frac{b^x}{\log (b)} \]

Antiderivative was successfully verified.

[In]

Integrate[b^x,x]

[Out]

b^x/Log[b]

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Maple [A]  time = 0.003, size = 9, normalized size = 1.1 \begin{align*}{\frac{{b}^{x}}{\ln \left ( b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(b^x,x)

[Out]

b^x/ln(b)

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Maxima [A]  time = 0.940985, size = 11, normalized size = 1.38 \begin{align*} \frac{b^{x}}{\log \left (b\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(b^x,x, algorithm="maxima")

[Out]

b^x/log(b)

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Fricas [A]  time = 1.62097, size = 16, normalized size = 2. \begin{align*} \frac{b^{x}}{\log \left (b\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(b^x,x, algorithm="fricas")

[Out]

b^x/log(b)

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Sympy [A]  time = 0.082876, size = 8, normalized size = 1. \begin{align*} \begin{cases} \frac{b^{x}}{\log{\left (b \right )}} & \text{for}\: \log{\left (b \right )} \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(b**x,x)

[Out]

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

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Giac [A]  time = 1.08596, size = 11, normalized size = 1.38 \begin{align*} \frac{b^{x}}{\log \left (b\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(b^x,x, algorithm="giac")

[Out]

b^x/log(b)