Optimal. Leaf size=22 \[ e^4 \text{ExpIntegralEi}(2 x-4)-e^2 \text{ExpIntegralEi}(2 x-2) \]
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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.
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Rule 2268
Rule 2178
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.
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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.
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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.
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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.
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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.
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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.
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