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

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

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Rubi [A]  time = 0.01, 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 = {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*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {a^{m x}}{m \log (a)}+x \]

Antiderivative was successfully verified.

[In]

Integrate[1 + a^(m*x),x]

[Out]

x + a^(m*x)/(m*Log[a])

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1+a^{m x}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[1 + a^(m*x),x]

[Out]

Could not integrate

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fricas [A]  time = 1.33, size = 19, normalized size = 1.27 \[ \frac {m x \log \relax (a) + a^{m x}}{m \log \relax (a)} \]

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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giac [A]  time = 0.58, size = 15, normalized size = 1.00 \[ x + \frac {a^{m x}}{m \log \relax (a)} \]

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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maple [A]  time = 0.04, size = 16, normalized size = 1.07

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

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 = 0.57, size = 15, normalized size = 1.00 \[ x + \frac {a^{m x}}{m \log \relax (a)} \]

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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mupad [B]  time = 0.32, size = 15, normalized size = 1.00 \[ x+\frac {a^{m\,x}}{m\,\ln \relax (a)} \]

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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sympy [A]  time = 0.09, size = 15, normalized size = 1.00 \[ x + \begin {cases} \frac {a^{m x}}{m \log {\relax (a )}} & \text {for}\: m \log {\relax (a )} \neq 0 \\x & \text {otherwise} \end {cases} \]

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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