3.2.58 \(\int e^{a x} x \, dx\) [158]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 = {2207, 2225} \begin {gather*} \frac {x e^{a x}}{a}-\frac {e^{a x}}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(a*x)/a^2) + (E^(a*x)*x)/a

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*} \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*}

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Mathematica [A]
time = 0.01, size = 14, normalized size = 0.67 \begin {gather*} \frac {e^{a x} (-1+a x)}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(a*x)*(-1 + a*x))/a^2

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 1.95, size = 27, normalized size = 1.29 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\left (-1+a x\right ) E^{a x}}{a^2},a^2\text {!=}0\right \}\right \},\frac {x^2}{2}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{(-1 + a x) E ^ (a x) / a ^ 2, a ^ 2 != 0}}, x ^ 2 / 2]

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Maple [A]
time = 0.01, size = 19, normalized size = 0.90

method	result	size
gosper	\(\frac {\left (a x -1\right ) {\mathrm e}^{a x}}{a^{2}}\)	\(14\)
risch	\(\frac {\left (a x -1\right ) {\mathrm e}^{a x}}{a^{2}}\)	\(14\)
derivativedivides	\(\frac {a \,{\mathrm e}^{a x} x -{\mathrm e}^{a x}}{a^{2}}\)	\(19\)
default	\(\frac {a \,{\mathrm e}^{a x} x -{\mathrm e}^{a x}}{a^{2}}\)	\(19\)
meijerg	\(\frac {1-\frac {\left (-2 a x +2\right ) {\mathrm e}^{a x}}{2}}{a^{2}}\)	\(19\)
norman	\(-\frac {{\mathrm e}^{a x}}{a^{2}}+\frac {{\mathrm e}^{a x} x}{a}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 13, normalized size = 0.62 \begin {gather*} \frac {{\left (a x - 1\right )} e^{\left (a x\right )}}{a^{2}} \end {gather*}

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 = 0.31, size = 13, normalized size = 0.62 \begin {gather*} \frac {{\left (a x - 1\right )} e^{\left (a x\right )}}{a^{2}} \end {gather*}

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 = 0.05, size = 19, normalized size = 0.90 \begin {gather*} \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 {gather*}

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 = 0.00, size = 14, normalized size = 0.67 \begin {gather*} \frac {\left (x a-1\right ) \mathrm {e}^{a x}}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x - 1)*e^(a*x)/a^2

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Mupad [B]
time = 0.03, size = 13, normalized size = 0.62 \begin {gather*} \frac {{\mathrm {e}}^{a\,x}\,\left (a\,x-1\right )}{a^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(a*x)*(a*x - 1))/a^2

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