3.174 \(\int \frac{e^{2 x}}{2-3 x+x^2} \, dx\)

Optimal. Leaf size=22 \[ e^4 \text{ExpIntegralEi}(2 x-4)-e^2 \text{ExpIntegralEi}(2 x-2) \]

[Out]

E^4*ExpIntegralEi[-4 + 2*x] - E^2*ExpIntegralEi[-2 + 2*x]

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Rubi [A]  time = 0.0559034, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2268, 2178} \[ e^4 \text{ExpIntegralEi}(2 x-4)-e^2 \text{ExpIntegralEi}(2 x-2) \]

Antiderivative was successfully verified.

[In]

Int[E^(2*x)/(2 - 3*x + x^2),x]

[Out]

E^4*ExpIntegralEi[-4 + 2*x] - E^2*ExpIntegralEi[-2 + 2*x]

Rule 2268

Int[(F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[F^(g*(d + e*x)^n), 1/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int \frac{e^{2 x}}{2-3 x+x^2} \, dx &=\int \left (-\frac{2 e^{2 x}}{4-2 x}-\frac{2 e^{2 x}}{-2+2 x}\right ) \, dx\\ &=-\left (2 \int \frac{e^{2 x}}{4-2 x} \, dx\right )-2 \int \frac{e^{2 x}}{-2+2 x} \, dx\\ &=e^4 \text{Ei}(-4+2 x)-e^2 \text{Ei}(-2+2 x)\\ \end{align*}

Mathematica [A]  time = 0.0325951, size = 22, normalized size = 1. \[ e^4 \text{ExpIntegralEi}(2 x-4)-e^2 \text{ExpIntegralEi}(2 x-2) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x)/(2 - 3*x + x^2),x]

[Out]

E^4*ExpIntegralEi[-4 + 2*x] - E^2*ExpIntegralEi[-2 + 2*x]

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Maple [A]  time = 0.006, size = 23, normalized size = 1.1 \begin{align*}{{\rm e}^{2}}{\it Ei} \left ( 1,2-2\,x \right ) -{{\rm e}^{4}}{\it Ei} \left ( 1,4-2\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)/(x^2-3*x+2),x)

[Out]

exp(2)*Ei(1,2-2*x)-exp(4)*Ei(1,4-2*x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (2 \, x\right )}}{x^{2} - 3 \, x + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)/(x^2-3*x+2),x, algorithm="maxima")

[Out]

integrate(e^(2*x)/(x^2 - 3*x + 2), x)

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Fricas [A]  time = 1.13443, size = 47, normalized size = 2.14 \begin{align*}{\rm Ei}\left (2 \, x - 4\right ) e^{4} -{\rm Ei}\left (2 \, x - 2\right ) e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)/(x^2-3*x+2),x, algorithm="fricas")

[Out]

Ei(2*x - 4)*e^4 - Ei(2*x - 2)*e^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{2 x}}{\left (x - 2\right ) \left (x - 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)/(x**2-3*x+2),x)

[Out]

Integral(exp(2*x)/((x - 2)*(x - 1)), x)

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Giac [A]  time = 1.09543, size = 27, normalized size = 1.23 \begin{align*}{\rm Ei}\left (2 \, x - 4\right ) e^{4} -{\rm Ei}\left (2 \, x - 2\right ) e^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2*x)/(x^2-3*x+2),x, algorithm="giac")

[Out]

Ei(2*x - 4)*e^4 - Ei(2*x - 2)*e^2