\(\int (1+a^{m x}) \, dx\) [512]

Optimal result

Integrand size = 7, antiderivative size = 15 \[ \int \left (1+a^{m x}\right ) \, dx=x+\frac {a^{m x}}{m \log (a)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2225} \[ \int \left (1+a^{m x}\right ) \, dx=\frac {a^{m x}}{m \log (a)}+x \]

[In]

Int[1 + a^(m*x),x]

[Out]

x + a^(m*x)/(m*Log[a])

Rule 2225

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (1+a^{m x}\right ) \, dx=x+\frac {a^{m x}}{m \log (a)} \]

[In]

Integrate[1 + a^(m*x),x]

[Out]

x + a^(m*x)/(m*Log[a])

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07

method	result	size
default	\(x +\frac {a^{m x}}{m \ln \left (a \right )}\)	\(16\)
risch	\(x +\frac {a^{m x}}{m \ln \left (a \right )}\)	\(16\)
parallelrisch	\(x +\frac {a^{m x}}{m \ln \left (a \right )}\)	\(16\)
parts	\(x +\frac {a^{m x}}{m \ln \left (a \right )}\)	\(16\)
norman	\(x +\frac {{\mathrm e}^{m x \ln \left (a \right )}}{m \ln \left (a \right )}\)	\(17\)
derivativedivides	\(\frac {a^{m x}+\ln \left (a^{m x}\right )}{m \ln \left (a \right )}\)	\(21\)

[In]

int(1+a^(m*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27 \[ \int \left (1+a^{m x}\right ) \, dx=\frac {m x \log \left (a\right ) + a^{m x}}{m \log \left (a\right )} \]

[In]

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

[Out]

(m*x*log(a) + a^(m*x))/(m*log(a))

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (1+a^{m x}\right ) \, dx=x + \begin {cases} \frac {a^{m x}}{m \log {\left (a \right )}} & \text {for}\: m \log {\left (a \right )} \neq 0 \\x & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (1+a^{m x}\right ) \, dx=x + \frac {a^{m x}}{m \log \left (a\right )} \]

[In]

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

[Out]

x + a^(m*x)/(m*log(a))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (1+a^{m x}\right ) \, dx=x + \frac {a^{m x}}{m \log \left (a\right )} \]

[In]

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

[Out]

x + a^(m*x)/(m*log(a))

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (1+a^{m x}\right ) \, dx=x+\frac {a^{m\,x}}{m\,\ln \left (a\right )} \]

[In]

int(a^(m*x) + 1,x)

[Out]

x + a^(m*x)/(m*log(a))