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

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = {2225} \begin {gather*} \frac {a^{m x}}{m \log (a)}+x \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} x+\frac {a^{m x}}{m \log (a)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + a^(m*x),x]

[Out]

x + a^(m*x)/(m*Log[a])

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Maple [A]
time = 0.02, size = 16, normalized size = 1.07

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 1.77, size = 15, normalized size = 1.00 \begin {gather*} x + \frac {a^{m x}}{m \log \left (a\right )} \end {gather*}

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))

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Fricas [A]
time = 0.47, size = 19, normalized size = 1.27 \begin {gather*} \frac {m x \log \left (a\right ) + a^{m x}}{m \log \left (a\right )} \end {gather*}

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))

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Sympy [A]
time = 0.02, size = 15, normalized size = 1.00 \begin {gather*} 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 {gather*}

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))

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Giac [A]
time = 1.15, size = 15, normalized size = 1.00 \begin {gather*} x + \frac {a^{m x}}{m \log \left (a\right )} \end {gather*}

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))

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Mupad [B]
time = 0.32, size = 15, normalized size = 1.00 \begin {gather*} x+\frac {a^{m\,x}}{m\,\ln \left (a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^(m*x) + 1,x)

[Out]

x + a^(m*x)/(m*log(a))

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