Optimal. Leaf size=62 \[ \frac{e^{\frac{b e}{b c-a d}} \text{Ei}\left (-\frac{d e (a+b x)}{(b c-a d) (c+d x)}\right )}{b}-\frac{\text{Ei}\left (\frac{e}{c+d x}\right )}{b} \]
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Rubi [A] time = 0.202567, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2222, 2210, 2228, 2178} \[ \frac{e^{\frac{b e}{b c-a d}} \text{Ei}\left (-\frac{d e (a+b x)}{(b c-a d) (c+d x)}\right )}{b}-\frac{\text{Ei}\left (\frac{e}{c+d x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2222
Rule 2210
Rule 2228
Rule 2178
Rubi steps
\begin{align*} \int \frac{e^{\frac{e}{c+d x}}}{a+b x} \, dx &=\frac{d \int \frac{e^{\frac{e}{c+d x}}}{c+d x} \, dx}{b}-\frac{(-b c+a d) \int \frac{e^{\frac{e}{c+d x}}}{(a+b x) (c+d x)} \, dx}{b}\\ &=-\frac{\text{Ei}\left (\frac{e}{c+d x}\right )}{b}+\frac{\operatorname{Subst}\left (\int \frac{\exp \left (-\frac{b e}{-b c+a d}+\frac{d e x}{-b c+a d}\right )}{x} \, dx,x,\frac{a+b x}{c+d x}\right )}{b}\\ &=-\frac{\text{Ei}\left (\frac{e}{c+d x}\right )}{b}+\frac{e^{\frac{b e}{b c-a d}} \text{Ei}\left (-\frac{d e (a+b x)}{(b c-a d) (c+d x)}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0625747, size = 56, normalized size = 0.9 \[ \frac{e^{\frac{b e}{b c-a d}} \text{Ei}\left (e \left (\frac{b}{a d-b c}+\frac{1}{c+d x}\right )\right )-\text{Ei}\left (\frac{e}{c+d x}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 79, normalized size = 1.3 \begin{align*} -{\frac{e}{d} \left ( -{\frac{d}{be}{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) }+{\frac{d}{be}{{\rm e}^{-{\frac{be}{ad-bc}}}}{\it Ei} \left ( 1,-{\frac{e}{dx+c}}-{\frac{be}{ad-bc}} \right ) } \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 (\frac{e}{d x + c}\right )}}{b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57333, size = 138, normalized size = 2.23 \begin{align*} \frac{{\rm Ei}\left (-\frac{b d e x + a d e}{b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac{b e}{b c - a d}\right )} -{\rm Ei}\left (\frac{e}{d x + c}\right )}{b} \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^{\frac{e}{c + d x}}}{a + b x}\, dx \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{e^{\left (\frac{e}{d x + c}\right )}}{b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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