Optimal. Leaf size=38 \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{\sqrt{b} \sqrt{\log (F)}} \]
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Rubi [A] time = 0.025179, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2180, 2204} \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{\sqrt{b} \sqrt{\log (F)}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \frac{F^{a+b x}}{\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int F^{a+b x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{\sqrt{b} \sqrt{\log (F)}}\\ \end{align*}
Mathematica [A] time = 0.0063854, size = 30, normalized size = 0.79 \[ -\frac{\sqrt{x} F^a \text{Gamma}\left (\frac{1}{2},-b x \log (F)\right )}{\sqrt{-b x \log (F)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 27, normalized size = 0.7 \begin{align*}{{F}^{a}\sqrt{\pi }{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ){\frac{1}{\sqrt{b}}}{\frac{1}{\sqrt{\ln \left ( F \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15337, size = 39, normalized size = 1.03 \begin{align*} \frac{\sqrt{\pi } F^{a} \sqrt{x}{\left (\operatorname{erf}\left (\sqrt{-b x \log \left (F\right )}\right ) - 1\right )}}{\sqrt{-b x \log \left (F\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5287, size = 96, normalized size = 2.53 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b \log \left (F\right )} F^{a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{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 \frac{F^{a + b x}}{\sqrt{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34626, size = 38, normalized size = 1. \begin{align*} -\frac{\sqrt{\pi } F^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{\sqrt{-b \log \left (F\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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