\(\int e^x x \, dx\) [74]

Optimal result

Integrand size = 5, antiderivative size = 11 \[ \int e^x x \, dx=-e^x+e^x x \]

[Out]

-exp(x)+exp(x)*x

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2207, 2225} \[ \int e^x x \, dx=e^x x-e^x \]

[In]

Int[E^x*x,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64 \[ \int e^x x \, dx=e^x (-1+x) \]

[In]

Integrate[E^x*x,x]

[Out]

E^x*(-1 + x)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64

method	result	size
gosper	\(\left (-1+x \right ) {\mathrm e}^{x}\)	\(7\)
risch	\(\left (-1+x \right ) {\mathrm e}^{x}\)	\(7\)
default	\(-{\mathrm e}^{x}+{\mathrm e}^{x} x\)	\(10\)
norman	\(-{\mathrm e}^{x}+{\mathrm e}^{x} x\)	\(10\)
parallelrisch	\(-{\mathrm e}^{x}+{\mathrm e}^{x} x\)	\(10\)
parts	\(-{\mathrm e}^{x}+{\mathrm e}^{x} x\)	\(10\)
meijerg	\(1-\frac {\left (-2 x +2\right ) {\mathrm e}^{x}}{2}\)	\(12\)

[In]

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

[Out]

(-1+x)*exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int e^x x \, dx={\left (x - 1\right )} e^{x} \]

[In]

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

[Out]

(x - 1)*e^x

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.45 \[ \int e^x x \, dx=\left (x - 1\right ) e^{x} \]

[In]

integrate(exp(x)*x,x)

[Out]

(x - 1)*exp(x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int e^x x \, dx={\left (x - 1\right )} e^{x} \]

[In]

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

[Out]

(x - 1)*e^x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int e^x x \, dx={\left (x - 1\right )} e^{x} \]

[In]

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

[Out]

(x - 1)*e^x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int e^x x \, dx={\mathrm {e}}^x\,\left (x-1\right ) \]

[In]

int(x*exp(x),x)

[Out]

exp(x)*(x - 1)