\(\int \frac {1}{2} (7+2 e^x) \, dx\) [4988]

Optimal result

Integrand size = 11, antiderivative size = 12 \[ \int \frac {1}{2} \left (7+2 e^x\right ) \, dx=-\frac {9}{4}+e^x+\frac {7 x}{2} \]

[Out]

7/2*x-9/4+exp(x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2225} \[ \int \frac {1}{2} \left (7+2 e^x\right ) \, dx=\frac {7 x}{2}+e^x \]

[In]

Int[(7 + 2*E^x)/2,x]

[Out]

E^x + (7*x)/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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}& = \frac {1}{2} \int \left (7+2 e^x\right ) \, dx \\ & = \frac {7 x}{2}+\int e^x \, dx \\ & = e^x+\frac {7 x}{2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {1}{2} \left (7+2 e^x\right ) \, dx=e^x+\frac {7 x}{2} \]

[In]

Integrate[(7 + 2*E^x)/2,x]

[Out]

E^x + (7*x)/2

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58

method	result	size
default	\(\frac {7 x}{2}+{\mathrm e}^{x}\)	\(7\)
norman	\(\frac {7 x}{2}+{\mathrm e}^{x}\)	\(7\)
risch	\(\frac {7 x}{2}+{\mathrm e}^{x}\)	\(7\)
parallelrisch	\(\frac {7 x}{2}+{\mathrm e}^{x}\)	\(7\)
parts	\(\frac {7 x}{2}+{\mathrm e}^{x}\)	\(7\)
derivativedivides	\({\mathrm e}^{x}+\frac {7 \ln \left ({\mathrm e}^{x}\right )}{2}\)	\(9\)

[In]

int(exp(x)+7/2,x,method=_RETURNVERBOSE)

[Out]

7/2*x+exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.50 \[ \int \frac {1}{2} \left (7+2 e^x\right ) \, dx=\frac {7}{2} \, x + e^{x} \]

[In]

integrate(exp(x)+7/2,x, algorithm="fricas")

[Out]

7/2*x + e^x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.58 \[ \int \frac {1}{2} \left (7+2 e^x\right ) \, dx=\frac {7 x}{2} + e^{x} \]

[In]

integrate(exp(x)+7/2,x)

[Out]

7*x/2 + exp(x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.50 \[ \int \frac {1}{2} \left (7+2 e^x\right ) \, dx=\frac {7}{2} \, x + e^{x} \]

[In]

integrate(exp(x)+7/2,x, algorithm="maxima")

[Out]

7/2*x + e^x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.50 \[ \int \frac {1}{2} \left (7+2 e^x\right ) \, dx=\frac {7}{2} \, x + e^{x} \]

[In]

integrate(exp(x)+7/2,x, algorithm="giac")

[Out]

7/2*x + e^x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.50 \[ \int \frac {1}{2} \left (7+2 e^x\right ) \, dx=\frac {7\,x}{2}+{\mathrm {e}}^x \]

[In]

int(exp(x) + 7/2,x)

[Out]

(7*x)/2 + exp(x)