Optimal. Leaf size=75 \[ \frac{(b+2 c x) f^{b x+c x^2}}{\log (f)}-\frac{\sqrt{\pi } \sqrt{c} f^{-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{\log ^{\frac{3}{2}}(f)} \]
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Rubi [A] time = 0.0575022, antiderivative size = 75, 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 = {2237, 2234, 2204} \[ \frac{(b+2 c x) f^{b x+c x^2}}{\log (f)}-\frac{\sqrt{\pi } \sqrt{c} f^{-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )}{\log ^{\frac{3}{2}}(f)} \]
Antiderivative was successfully verified.
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Rule 2237
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int f^{b x+c x^2} (b+2 c x)^2 \, dx &=\frac{f^{b x+c x^2} (b+2 c x)}{\log (f)}-\frac{(2 c) \int f^{b x+c x^2} \, dx}{\log (f)}\\ &=\frac{f^{b x+c x^2} (b+2 c x)}{\log (f)}-\frac{\left (2 c f^{-\frac{b^2}{4 c}}\right ) \int f^{\frac{(b+2 c x)^2}{4 c}} \, dx}{\log (f)}\\ &=-\frac{\sqrt{c} f^{-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{(b+2 c x) \sqrt{\log (f)}}{2 \sqrt{c}}\right )}{\log ^{\frac{3}{2}}(f)}+\frac{f^{b x+c x^2} (b+2 c x)}{\log (f)}\\ \end{align*}
Mathematica [A] time = 0.0619255, size = 84, normalized size = 1.12 \[ \frac{f^{-\frac{b^2}{4 c}} \left (\sqrt{\log (f)} (b+2 c x) f^{\frac{(b+2 c x)^2}{4 c}}-\sqrt{\pi } \sqrt{c} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )\right )}{\log ^{\frac{3}{2}}(f)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 90, normalized size = 1.2 \begin{align*} 2\,{\frac{cx{f}^{c{x}^{2}}{f}^{bx}}{\ln \left ( f \right ) }}+{\frac{b{f}^{c{x}^{2}}{f}^{bx}}{\ln \left ( f \right ) }}+{\frac{c\sqrt{\pi }}{\ln \left ( f \right ) }{f}^{-{\frac{{b}^{2}}{4\,c}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) }x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.24178, size = 444, normalized size = 5.92 \begin{align*} \frac{\sqrt{\pi } b^{2} \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} x - \frac{b \log \left (f\right )}{2 \, \sqrt{-c \log \left (f\right )}}\right )}{2 \, \sqrt{-c \log \left (f\right )} f^{\frac{b^{2}}{4 \, c}}} - \frac{{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b{\left (\operatorname{erf}\left (\frac{1}{2} \, \sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{2}}{\sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac{3}{2}}} - \frac{2 \, c f^{\frac{{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )}{\left (c \log \left (f\right )\right )^{\frac{3}{2}}}\right )} b c}{\sqrt{c \log \left (f\right )} f^{\frac{b^{2}}{4 \, c}}} + \frac{{\left (\frac{\sqrt{\pi }{\left (2 \, c x + b\right )} b^{2}{\left (\operatorname{erf}\left (\frac{1}{2} \, \sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{3}}{\sqrt{-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac{5}{2}}} - \frac{4 \,{\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac{3}{2}, -\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{4 \, c}\right ) \log \left (f\right )^{3}}{\left (-\frac{{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}\right )^{\frac{3}{2}} \left (c \log \left (f\right )\right )^{\frac{5}{2}}} - \frac{4 \, b c f^{\frac{{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )^{2}}{\left (c \log \left (f\right )\right )^{\frac{5}{2}}}\right )} c^{2}}{2 \, \sqrt{c \log \left (f\right )} f^{\frac{b^{2}}{4 \, c}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54022, size = 171, normalized size = 2.28 \begin{align*} \frac{{\left (2 \, c x + b\right )} f^{c x^{2} + b x} \log \left (f\right ) + \frac{\sqrt{\pi } \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}}}}{\log \left (f\right )^{2}} \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 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 [A] time = 1.23602, size = 104, normalized size = 1.39 \begin{align*} \frac{c{\left (2 \, x + \frac{b}{c}\right )} e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right )\right )}}{\log \left (f\right )} + \frac{\sqrt{\pi } c \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right )}{\left (2 \, x + \frac{b}{c}\right )}\right )}{\sqrt{-c \log \left (f\right )} f^{\frac{b^{2}}{4 \, c}} \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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