Optimal. Leaf size=66 \[ \frac{\log (f) f^{-\frac{b^2}{4 c}} \text{Ei}\left (\frac{(b+2 c x)^2 \log (f)}{4 c}\right )}{16 c^2}-\frac{f^{b x+c x^2}}{4 c (b+2 c x)^2} \]
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Rubi [A] time = 0.0632942, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2239, 2238} \[ \frac{\log (f) f^{-\frac{b^2}{4 c}} \text{Ei}\left (\frac{(b+2 c x)^2 \log (f)}{4 c}\right )}{16 c^2}-\frac{f^{b x+c x^2}}{4 c (b+2 c x)^2} \]
Antiderivative was successfully verified.
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Rule 2239
Rule 2238
Rubi steps
\begin{align*} \int \frac{f^{b x+c x^2}}{(b+2 c x)^3} \, dx &=-\frac{f^{b x+c x^2}}{4 c (b+2 c x)^2}+\frac{\log (f) \int \frac{f^{b x+c x^2}}{b+2 c x} \, dx}{4 c}\\ &=-\frac{f^{b x+c x^2}}{4 c (b+2 c x)^2}+\frac{f^{-\frac{b^2}{4 c}} \text{Ei}\left (\frac{(b+2 c x)^2 \log (f)}{4 c}\right ) \log (f)}{16 c^2}\\ \end{align*}
Mathematica [A] time = 0.0494255, size = 77, normalized size = 1.17 \[ \frac{f^{-\frac{b^2}{4 c}} \left (\log (f) (b+2 c x)^2 \text{Ei}\left (\frac{(b+2 c x)^2 \log (f)}{4 c}\right )-4 c f^{\frac{(b+2 c x)^2}{4 c}}\right )}{16 c^2 (b+2 c x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 74, normalized size = 1.1 \begin{align*} -{\frac{1}{4\,c \left ( 2\,cx+b \right ) ^{2}}{f}^{{\frac{ \left ( 2\,cx+b \right ) ^{2}}{4\,c}}}{f}^{-{\frac{{b}^{2}}{4\,c}}}}-{\frac{\ln \left ( f \right ) }{16\,{c}^{2}}{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}}{{\left (2 \, c x + b\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52955, size = 215, normalized size = 3.26 \begin{align*} -\frac{4 \, c f^{c x^{2} + b x} - \frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )}{\rm Ei}\left (\frac{{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (f\right )}{4 \, c}\right ) \log \left (f\right )}{f^{\frac{b^{2}}{4 \, c}}}}{16 \,{\left (4 \, c^{4} x^{2} + 4 \, b c^{3} x + b^{2} c^{2}\right )}} \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}}}{\left (b + 2 c x\right )^{3}}\, 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}}{{\left (2 \, c x + b\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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