Optimal. Leaf size=81 \[ \frac{\sqrt{\pi } \sqrt{\log (f)} f^{-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2}}-\frac{f^{b x+c x^2}}{2 c (b+2 c x)} \]
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Rubi [A] time = 0.0485183, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2239, 2234, 2204} \[ \frac{\sqrt{\pi } \sqrt{\log (f)} f^{-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{4 c^{3/2}}-\frac{f^{b x+c x^2}}{2 c (b+2 c x)} \]
Antiderivative was successfully verified.
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Rule 2239
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int \frac{f^{b x+c x^2}}{(b+2 c x)^2} \, dx &=-\frac{f^{b x+c x^2}}{2 c (b+2 c x)}+\frac{\log (f) \int f^{b x+c x^2} \, dx}{2 c}\\ &=-\frac{f^{b x+c x^2}}{2 c (b+2 c x)}+\frac{\left (f^{-\frac{b^2}{4 c}} \log (f)\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{2 c}\\ &=-\frac{f^{b x+c x^2}}{2 c (b+2 c x)}+\frac{f^{-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right ) \sqrt{\log (f)}}{4 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0589249, size = 94, normalized size = 1.16 \[ \frac{f^{-\frac{b^2}{4 c}} \left (\sqrt{\pi } \sqrt{\log (f)} (b+2 c x) \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )-2 \sqrt{c} f^{\frac{(b+2 c x)^2}{4 c}}\right )}{4 c^{3/2} (b+2 c x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 87, normalized size = 1.1 \begin{align*} -{\frac{1}{2\,c \left ( 2\,cx+b \right ) }{f}^{{\frac{ \left ( 2\,cx+b \right ) ^{2}}{4\,c}}}{f}^{-{\frac{{b}^{2}}{4\,c}}}}+{\frac{\ln \left ( f \right ) \sqrt{\pi }}{4\,{c}^{2}}{f}^{-{\frac{{b}^{2}}{4\,c}}}{\it Erf} \left ({\frac{2\,cx+b}{2}\sqrt{-{\frac{\ln \left ( f \right ) }{c}}}} \right ){\frac{1}{\sqrt{-{\frac{\ln \left ( f \right ) }{c}}}}}} \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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62962, size = 186, normalized size = 2.3 \begin{align*} -\frac{2 \, c f^{c x^{2} + b x} + \frac{\sqrt{\pi }{\left (2 \, c x + b\right )} \sqrt{-c \log \left (f\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac{b^{2}}{4 \, c}}}}{4 \,{\left (2 \, c^{3} x + b 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 )^{2}}\, 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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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