Optimal. Leaf size=37 \[ \frac{f^{-\frac{b^2}{4 c}} \text{Ei}\left (\frac{(b+2 c x)^2 \log (f)}{4 c}\right )}{4 c} \]
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Rubi [A] time = 0.0294125, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {2238} \[ \frac{f^{-\frac{b^2}{4 c}} \text{Ei}\left (\frac{(b+2 c x)^2 \log (f)}{4 c}\right )}{4 c} \]
Antiderivative was successfully verified.
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Rule 2238
Rubi steps
\begin{align*} \int \frac{f^{b x+c x^2}}{b+2 c x} \, dx &=\frac{f^{-\frac{b^2}{4 c}} \text{Ei}\left (\frac{(b+2 c x)^2 \log (f)}{4 c}\right )}{4 c}\\ \end{align*}
Mathematica [A] time = 0.0332878, size = 37, normalized size = 1. \[ \frac{f^{-\frac{b^2}{4 c}} \text{Ei}\left (\frac{(b+2 c x)^2 \log (f)}{4 c}\right )}{4 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 33, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,c}{f}^{-{\frac{{b}^{2}}{4\,c}}}{\it Ei} \left ( 1,-{\frac{ \left ( 2\,cx+b \right ) ^{2}\ln \left ( f \right ) }{4\,c}} \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{f^{c x^{2} + b x}}{2 \, c x + b}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56193, size = 92, normalized size = 2.49 \begin{align*} \frac{{\rm Ei}\left (\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (f\right )}{4 \, c}\right )}{4 \, c f^{\frac{b^{2}}{4 \, c}}} \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{f^{b x + c x^{2}}}{b + 2 c 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{f^{c x^{2} + b x}}{2 \, c x + b}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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