\(\int e^x (1+x) \, dx\) [7996]

Optimal result

Integrand size = 7, antiderivative size = 7 \[ \int e^x (1+x) \, dx=-9+e^x x \]

[Out]

-9+exp(x)*x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.86, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2207, 2225} \[ \int e^x (1+x) \, dx=e^x (x+1)-e^x \]

[In]

Int[E^x*(1 + x),x]

[Out]

-E^x + E^x*(1 + x)

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

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}& = e^x (1+x)-\int e^x \, dx \\ & = -e^x+e^x (1+x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71 \[ \int e^x (1+x) \, dx=e^x x \]

[In]

Integrate[E^x*(1 + x),x]

[Out]

E^x*x

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71

method	result	size
gosper	\({\mathrm e}^{x} x\)	\(5\)
default	\({\mathrm e}^{x} x\)	\(5\)
norman	\({\mathrm e}^{x} x\)	\(5\)
risch	\({\mathrm e}^{x} x\)	\(5\)
parallelrisch	\({\mathrm e}^{x} x\)	\(5\)
parts	\({\mathrm e}^{x} x\)	\(5\)
meijerg	\({\mathrm e}^{x}-\frac {\left (2-2 x \right ) {\mathrm e}^{x}}{2}\)	\(13\)

[In]

int((1+x)*exp(x),x,method=_RETURNVERBOSE)

[Out]

exp(x)*x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.57 \[ \int e^x (1+x) \, dx=x e^{x} \]

[In]

integrate((1+x)*exp(x),x, algorithm="fricas")

[Out]

x*e^x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.43 \[ \int e^x (1+x) \, dx=x e^{x} \]

[In]

integrate((1+x)*exp(x),x)

[Out]

x*exp(x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.29 \[ \int e^x (1+x) \, dx={\left (x - 1\right )} e^{x} + e^{x} \]

[In]

integrate((1+x)*exp(x),x, algorithm="maxima")

[Out]

(x - 1)*e^x + e^x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.57 \[ \int e^x (1+x) \, dx=x e^{x} \]

[In]

integrate((1+x)*exp(x),x, algorithm="giac")

[Out]

x*e^x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.57 \[ \int e^x (1+x) \, dx=x\,{\mathrm {e}}^x \]

[In]

int(exp(x)*(x + 1),x)

[Out]

x*exp(x)