3.1.20 \(\int e^{2 x} x \, dx\) [20]

Optimal. Leaf size=20 \[ -\frac {e^{2 x}}{4}+\frac {1}{2} e^{2 x} x \]

[Out]

-1/4*exp(2*x)+1/2*exp(2*x)*x

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Rubi [A]
time = 0.00, antiderivative size = 20, 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 {1}{2} e^{2 x} x-\frac {e^{2 x}}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*x)*x,x]

[Out]

-1/4*E^(2*x) + (E^(2*x)*x)/2

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^{2 x} x \, dx &=\frac {1}{2} e^{2 x} x-\frac {1}{2} \int e^{2 x} \, dx\\ &=-\frac {e^{2 x}}{4}+\frac {1}{2} e^{2 x} x\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 15, normalized size = 0.75 \begin {gather*} e^{2 x} \left (-\frac {1}{4}+\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x)*x,x]

[Out]

E^(2*x)*(-1/4 + x/2)

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Mathics [A]
time = 1.76, size = 12, normalized size = 0.60 \begin {gather*} \frac {\left (-1+2 x\right ) E^{2 x}}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x*E^(2*x),x]')

[Out]

(-1 + 2 x) E ^ (2 x) / 4

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Maple [A]
time = 0.01, size = 15, normalized size = 0.75

method	result	size
risch	\(\left (-\frac {1}{4}+\frac {x}{2}\right ) {\mathrm e}^{2 x}\)	\(11\)
gosper	\(\frac {\left (2 x -1\right ) {\mathrm e}^{2 x}}{4}\)	\(12\)
meijerg	\(\frac {1}{4}-\frac {\left (2-4 x \right ) {\mathrm e}^{2 x}}{8}\)	\(14\)
derivativedivides	\(-\frac {{\mathrm e}^{2 x}}{4}+\frac {{\mathrm e}^{2 x} x}{2}\)	\(15\)
default	\(-\frac {{\mathrm e}^{2 x}}{4}+\frac {{\mathrm e}^{2 x} x}{2}\)	\(15\)
norman	\(-\frac {{\mathrm e}^{2 x}}{4}+\frac {{\mathrm e}^{2 x} x}{2}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)*x,x,method=_RETURNVERBOSE)

[Out]

-1/4*exp(2*x)+1/2*exp(2*x)*x

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Maxima [A]
time = 0.26, size = 11, normalized size = 0.55 \begin {gather*} \frac {1}{4} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)*x,x, algorithm="maxima")

[Out]

1/4*(2*x - 1)*e^(2*x)

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Fricas [A]
time = 0.32, size = 11, normalized size = 0.55 \begin {gather*} \frac {1}{4} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)*x,x, algorithm="fricas")

[Out]

1/4*(2*x - 1)*e^(2*x)

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Sympy [A]
time = 0.04, size = 10, normalized size = 0.50 \begin {gather*} \frac {\left (2 x - 1\right ) e^{2 x}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)*x,x)

[Out]

(2*x - 1)*exp(2*x)/4

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Giac [A]
time = 0.00, size = 12, normalized size = 0.60 \begin {gather*} \frac {1}{4} \left (2 x-1\right ) \mathrm {e}^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)*x,x)

[Out]

1/4*e^(2*x)*(2*x - 1)

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Mupad [B]
time = 0.02, size = 11, normalized size = 0.55 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}\,\left (2\,x-1\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*exp(2*x),x)

[Out]

(exp(2*x)*(2*x - 1))/4

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