Optimal. Leaf size=105 \[ \frac{\sqrt{d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac{\sqrt{\pi } \sqrt{e} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac{3}{2}}(F)} \]
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Rubi [A] time = 0.0772365, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2176, 2180, 2204} \[ \frac{\sqrt{d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac{\sqrt{\pi } \sqrt{e} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac{3}{2}}(F)} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int F^{c (a+b x)} \sqrt{d+e x} \, dx &=\frac{F^{c (a+b x)} \sqrt{d+e x}}{b c \log (F)}-\frac{e \int \frac{F^{c (a+b x)}}{\sqrt{d+e x}} \, dx}{2 b c \log (F)}\\ &=\frac{F^{c (a+b x)} \sqrt{d+e x}}{b c \log (F)}-\frac{\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 )}{b c \log (F)}\\ &=-\frac{\sqrt{e} 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 )}{2 b^{3/2} c^{3/2} \log ^{\frac{3}{2}}(F)}+\frac{F^{c (a+b x)} \sqrt{d+e x}}{b c \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0380728, size = 63, normalized size = 0.6 \[ -\frac{(d+e x)^{3/2} F^{c \left (a-\frac{b d}{e}\right )} \text{Gamma}\left (\frac{3}{2},-\frac{b c \log (F) (d+e x)}{e}\right )}{e \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) }\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 \sqrt{e x + d} F^{{\left (b x + a\right )} c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08921, size = 220, normalized size = 2.1 \begin{align*} \frac{2 \, \sqrt{e x + d} F^{b c x + a c} b c \log \left (F\right ) + \frac{\sqrt{\pi } \sqrt{-\frac{b c \log \left (F\right )}{e}} 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}}}}{2 \, b^{2} c^{2} \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^{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 = 1.30169, size = 170, normalized size = 1.62 \begin{align*} \frac{1}{2} \,{\left (\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 )} + 2\right )}}{\sqrt{-b c e \log \left (F\right )} b c \log \left (F\right )} + \frac{2 \, \sqrt{x e + d} e^{\left ({\left ({\left (x e + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right )\right )} e^{\left (-1\right )} + 1\right )}}{b c \log \left (F\right )}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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