3.20 \(\int e^{2 x} x \, dx\)

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

[Out]

-E^(2*x)/4 + (E^(2*x)*x)/2

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Rubi [A]  time = 0.0084638, antiderivative size = 20, 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} \[ \frac{1}{2} e^{2 x} x-\frac{e^{2 x}}{4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-E^(2*x)/4 + (E^(2*x)*x)/2

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^{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*}

Mathematica [A]  time = 0.0042397, size = 15, normalized size = 0.75 \[ e^{2 x} \left (\frac{x}{2}-\frac{1}{4}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(2*x)*(-1/4 + x/2)

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Maple [A]  time = 0., size = 12, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,x-1 \right ){{\rm e}^{2\,x}}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.936104, size = 15, normalized size = 0.75 \begin{align*} \frac{1}{4} \,{\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \end{align*}

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 = 1.8314, size = 31, normalized size = 1.55 \begin{align*} \frac{1}{4} \,{\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \end{align*}

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 = 0.075245, size = 10, normalized size = 0.5 \begin{align*} \frac{\left (2 x - 1\right ) e^{2 x}}{4} \end{align*}

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 = 1.05513, size = 15, normalized size = 0.75 \begin{align*} \frac{1}{4} \,{\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*x - 1)*e^(2*x)