Optimal. Leaf size=11 \[ \text{Ei}\left (a+b x+c x^2\right ) \]
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Rubi [A] time = 0.176594, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {6707, 2178} \[ \text{Ei}\left (a+b x+c x^2\right ) \]
Antiderivative was successfully verified.
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Rule 6707
Rule 2178
Rubi steps
\begin{align*} \int \frac{e^{a+b x+c x^2} (b+2 c x)}{a+b x+c x^2} \, dx &=\operatorname{Subst}\left (\int \frac{e^x}{x} \, dx,x,a+b x+c x^2\right )\\ &=\text{Ei}\left (a+b x+c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0237218, size = 10, normalized size = 0.91 \[ \text{Ei}(a+x (b+c x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 19, normalized size = 1.7 \begin{align*} -{\it Ei} \left ( 1,-c{x}^{2}-bx-a \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{{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{c x^{2} + b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.806188, size = 28, normalized size = 2.55 \begin{align*}{\rm Ei}\left (c x^{2} + b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 34.1153, size = 10, normalized size = 0.91 \begin{align*} \operatorname{Ei}{\left (a + b x + c x^{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{c x^{2} + b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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