3.512 \(\int (1+a^{m x}) \, dx\)

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

[Out]

x + a^(m*x)/(m*Log[a])

________________________________________________________________________________________

Rubi [A]  time = 0.0055083, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2194} \[ \frac{a^{m x}}{m \log (a)}+x \]

Antiderivative was successfully verified.

[In]

Int[1 + a^(m*x),x]

[Out]

x + a^(m*x)/(m*Log[a])

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 (1+a^{m x}\right ) \, dx &=x+\int a^{m x} \, dx\\ &=x+\frac{a^{m x}}{m \log (a)}\\ \end{align*}

Mathematica [A]  time = 0.0038371, size = 15, normalized size = 1. \[ \frac{a^{m x}}{m \log (a)}+x \]

Antiderivative was successfully verified.

[In]

Integrate[1 + a^(m*x),x]

[Out]

x + a^(m*x)/(m*Log[a])

________________________________________________________________________________________

Maple [A]  time = 0., size = 16, normalized size = 1.1 \begin{align*} x+{\frac{{a}^{mx}}{m\ln \left ( a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1+a^(m*x),x)

[Out]

x+a^(m*x)/m/ln(a)

________________________________________________________________________________________

Maxima [A]  time = 0.923701, size = 20, normalized size = 1.33 \begin{align*} x + \frac{a^{m x}}{m \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1+a^(m*x),x, algorithm="maxima")

[Out]

x + a^(m*x)/(m*log(a))

________________________________________________________________________________________

Fricas [A]  time = 1.82672, size = 47, normalized size = 3.13 \begin{align*} \frac{m x \log \left (a\right ) + a^{m x}}{m \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1+a^(m*x),x, algorithm="fricas")

[Out]

(m*x*log(a) + a^(m*x))/(m*log(a))

________________________________________________________________________________________

Sympy [A]  time = 0.087906, size = 15, normalized size = 1. \begin{align*} x + \begin{cases} \frac{a^{m x}}{m \log{\left (a \right )}} & \text{for}\: m \log{\left (a \right )} \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1+a**(m*x),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.12403, size = 20, normalized size = 1.33 \begin{align*} x + \frac{a^{m x}}{m \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1+a^(m*x),x, algorithm="giac")

[Out]

x + a^(m*x)/(m*log(a))