Optimal. Leaf size=41 \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b \sqrt{c} \sqrt{\log (f)}} \]
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Rubi [A] time = 0.0078993, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2204} \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b \sqrt{c} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rubi steps
\begin{align*} \int f^{c (a+b x)^2} \, dx &=\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{c} (a+b x) \sqrt{\log (f)}\right )}{2 b \sqrt{c} \sqrt{\log (f)}}\\ \end{align*}
Mathematica [A] time = 0.0038041, size = 41, normalized size = 1. \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )}{2 b \sqrt{c} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 41, normalized size = 1. \begin{align*} -{\frac{\sqrt{\pi }}{2\,b}{\it Erf} \left ( -b\sqrt{-c\ln \left ( f \right ) }x+{ac\ln \left ( f \right ){\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 [A] time = 1.14094, size = 54, normalized size = 1.32 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{-c \log \left (f\right )} b x - \frac{a c \log \left (f\right )}{\sqrt{-c \log \left (f\right )}}\right )}{2 \, \sqrt{-c \log \left (f\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55447, size = 117, normalized size = 2.85 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b^{2} c \log \left (f\right )} \operatorname{erf}\left (\frac{\sqrt{-b^{2} c \log \left (f\right )}{\left (b x + a\right )}}{b}\right )}{2 \, b^{2} c \log \left (f\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 f^{c \left (a + b x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20551, size = 45, normalized size = 1.1 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c \log \left (f\right )} b{\left (x + \frac{a}{b}\right )}\right )}{2 \, \sqrt{-c \log \left (f\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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