Optimal. Leaf size=37 \[ \frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{2 \sqrt{b} \sqrt{\log (f)}} \]
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Rubi [A] time = 0.006949, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2204} \[ \frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{2 \sqrt{b} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Rule 2204
Rubi steps
\begin{align*} \int f^{a+b x^2} \, dx &=\frac{f^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{2 \sqrt{b} \sqrt{\log (f)}}\\ \end{align*}
Mathematica [A] time = 0.0045511, size = 37, normalized size = 1. \[ \frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )}{2 \sqrt{b} \sqrt{\log (f)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 26, normalized size = 0.7 \begin{align*}{\frac{{f}^{a}\sqrt{\pi }}{2}{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06032, size = 34, normalized size = 0.92 \begin{align*} \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{2 \, \sqrt{-b \log \left (f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78363, size = 93, normalized size = 2.51 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b \log \left (f\right )} f^{a} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right )}{2 \, b \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^{a + b x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22289, size = 35, normalized size = 0.95 \begin{align*} -\frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (f\right )} x\right )}{2 \, \sqrt{-b \log \left (f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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