Optimal. Leaf size=97 \[ \frac{2 \sqrt{\pi } \sqrt{b} \sqrt{c} \sqrt{\log (F)} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{e^{3/2}}-\frac{2 F^{c (a+b x)}}{e \sqrt{d+e x}} \]
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Rubi [A] time = 0.0854085, antiderivative size = 97, 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 = {2177, 2180, 2204} \[ \frac{2 \sqrt{\pi } \sqrt{b} \sqrt{c} \sqrt{\log (F)} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{e^{3/2}}-\frac{2 F^{c (a+b x)}}{e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \frac{F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx &=-\frac{2 F^{c (a+b x)}}{e \sqrt{d+e x}}+\frac{(2 b c \log (F)) \int \frac{F^{c (a+b x)}}{\sqrt{d+e x}} \, dx}{e}\\ &=-\frac{2 F^{c (a+b x)}}{e \sqrt{d+e x}}+\frac{(4 b c \log (F)) \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^2}\\ &=-\frac{2 F^{c (a+b x)}}{e \sqrt{d+e x}}+\frac{2 \sqrt{b} \sqrt{c} 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{\log (F)}}{e^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0744716, size = 75, normalized size = 0.77 \[ -\frac{2 \left (F^{c (a+b x)}-F^{c \left (a-\frac{b d}{e}\right )} \sqrt{-\frac{b c \log (F) (d+e x)}{e}} \text{Gamma}\left (\frac{1}{2},-\frac{b c \log (F) (d+e x)}{e}\right )\right )}{e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{{F}^{c \left ( bx+a \right ) } \left ( ex+d \right ) ^{-{\frac{3}{2}}}}\, 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}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83572, size = 205, normalized size = 2.11 \begin{align*} -\frac{2 \,{\left (\frac{\sqrt{\pi }{\left (e x + d\right )} \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}}} + \sqrt{e x + d} F^{b c x + a c}\right )}}{e^{2} x + d e} \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 )}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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