Optimal. Leaf size=72 \[ \frac{\sqrt{\pi } F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{\sqrt{b} \sqrt{c} \sqrt{e} \sqrt{\log (F)}} \]
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Rubi [A] time = 0.0454783, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2180, 2204} \[ \frac{\sqrt{\pi } F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{\sqrt{b} \sqrt{c} \sqrt{e} \sqrt{\log (F)}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \frac{F^{c (a+b x)}}{\sqrt{d+e x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int F^{c \left (a-\frac{b d}{e}\right )+\frac{b c x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{e}\\ &=\frac{F^{c \left (a-\frac{b d}{e}\right )} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{d+e x} \sqrt{\log (F)}}{\sqrt{e}}\right )}{\sqrt{b} \sqrt{c} \sqrt{e} \sqrt{\log (F)}}\\ \end{align*}
Mathematica [A] time = 0.0279241, size = 63, normalized size = 0.88 \[ -\frac{\sqrt{d+e x} F^{c \left (a-\frac{b d}{e}\right )} \text{Gamma}\left (\frac{1}{2},-\frac{b c \log (F) (d+e x)}{e}\right )}{e \sqrt{-\frac{b c \log (F) (d+e x)}{e}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.022, size = 0, normalized size = 0. \begin{align*} \int{{F}^{c \left ( bx+a \right ) }{\frac{1}{\sqrt{ex+d}}}}\, dx \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^{{\left (b x + a\right )} c}}{\sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04574, size = 142, normalized size = 1.97 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-\frac{b c \log \left (F\right )}{e}} \operatorname{erf}\left (\sqrt{e x + d} \sqrt{-\frac{b c \log \left (F\right )}{e}}\right )}{F^{\frac{b c d - a c e}{e}} b 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 \frac{F^{c \left (a + b x\right )}}{\sqrt{d + e x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25441, size = 78, normalized size = 1.08 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-b c e \log \left (F\right )} \sqrt{x e + d} e^{\left (-1\right )}\right ) e^{\left (-{\left (b c d \log \left (F\right ) - a c e \log \left (F\right )\right )} e^{\left (-1\right )}\right )}}{\sqrt{-b c e \log \left (F\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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