3.158 \(\int e^{a x} x \, dx\)

Optimal. Leaf size=21 \[ \frac{x e^{a x}}{a}-\frac{e^{a x}}{a^2} \]

[Out]

-(E^(a*x)/a^2) + (E^(a*x)*x)/a

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Rubi [A]  time = 0.0086599, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2176, 2194} \[ \frac{x e^{a x}}{a}-\frac{e^{a x}}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(a*x)*x,x]

[Out]

-(E^(a*x)/a^2) + (E^(a*x)*x)/a

Rule 2176

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] &&  !$UseGamma === True

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 e^{a x} x \, dx &=\frac{e^{a x} x}{a}-\frac{\int e^{a x} \, dx}{a}\\ &=-\frac{e^{a x}}{a^2}+\frac{e^{a x} x}{a}\\ \end{align*}

Mathematica [A]  time = 0.0056161, size = 14, normalized size = 0.67 \[ \frac{e^{a x} (a x-1)}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(a*x)*x,x]

[Out]

(E^(a*x)*(-1 + a*x))/a^2

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Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \begin{align*}{\frac{ \left ( ax-1 \right ){{\rm e}^{ax}}}{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a*x)*x,x)

[Out]

(a*x-1)*exp(a*x)/a^2

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Maxima [A]  time = 0.942147, size = 18, normalized size = 0.86 \begin{align*} \frac{{\left (a x - 1\right )} e^{\left (a x\right )}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x - 1)*e^(a*x)/a^2

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Fricas [A]  time = 1.6796, size = 31, normalized size = 1.48 \begin{align*} \frac{{\left (a x - 1\right )} e^{\left (a x\right )}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x - 1)*e^(a*x)/a^2

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Sympy [A]  time = 0.101134, size = 19, normalized size = 0.9 \begin{align*} \begin{cases} \frac{\left (a x - 1\right ) e^{a x}}{a^{2}} & \text{for}\: a^{2} \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(a*x)*x,x)

[Out]

Piecewise(((a*x - 1)*exp(a*x)/a**2, Ne(a**2, 0)), (x**2/2, True))

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Giac [A]  time = 1.09533, size = 18, normalized size = 0.86 \begin{align*} \frac{{\left (a x - 1\right )} e^{\left (a x\right )}}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x - 1)*e^(a*x)/a^2