Optimal. Leaf size=138 \[ \frac{3 \sqrt{\pi } e^{3/2} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{4 b^{5/2} c^{5/2} \log ^{\frac{5}{2}}(F)}-\frac{3 e \sqrt{d+e x} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac{(d+e x)^{3/2} F^{c (a+b x)}}{b c \log (F)} \]
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Rubi [A] time = 0.11937, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2176, 2180, 2204} \[ \frac{3 \sqrt{\pi } e^{3/2} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{4 b^{5/2} c^{5/2} \log ^{\frac{5}{2}}(F)}-\frac{3 e \sqrt{d+e x} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac{(d+e x)^{3/2} F^{c (a+b x)}}{b c \log (F)} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int F^{c (a+b x)} (d+e x)^{3/2} \, dx &=\frac{F^{c (a+b x)} (d+e x)^{3/2}}{b c \log (F)}-\frac{(3 e) \int F^{c (a+b x)} \sqrt{d+e x} \, dx}{2 b c \log (F)}\\ &=-\frac{3 e F^{c (a+b x)} \sqrt{d+e x}}{2 b^2 c^2 \log ^2(F)}+\frac{F^{c (a+b x)} (d+e x)^{3/2}}{b c \log (F)}+\frac{\left (3 e^2\right ) \int \frac{F^{c (a+b x)}}{\sqrt{d+e x}} \, dx}{4 b^2 c^2 \log ^2(F)}\\ &=-\frac{3 e F^{c (a+b x)} \sqrt{d+e x}}{2 b^2 c^2 \log ^2(F)}+\frac{F^{c (a+b x)} (d+e x)^{3/2}}{b c \log (F)}+\frac{(3 e) \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 )}{2 b^2 c^2 \log ^2(F)}\\ &=\frac{3 e^{3/2} 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 )}{4 b^{5/2} c^{5/2} \log ^{\frac{5}{2}}(F)}-\frac{3 e F^{c (a+b x)} \sqrt{d+e x}}{2 b^2 c^2 \log ^2(F)}+\frac{F^{c (a+b x)} (d+e x)^{3/2}}{b c \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0700684, size = 63, normalized size = 0.46 \[ -\frac{(d+e x)^{5/2} F^{c \left (a-\frac{b d}{e}\right )} \text{Gamma}\left (\frac{5}{2},-\frac{b c \log (F) (d+e x)}{e}\right )}{e \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{5/2}} \]
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{\left (e x + d\right )}^{\frac{3}{2}} 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 = 1.91564, size = 288, normalized size = 2.09 \begin{align*} -\frac{\frac{3 \, \sqrt{\pi } \sqrt{-\frac{b c \log \left (F\right )}{e}} e^{2} \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 \,{\left (3 \, b c e \log \left (F\right ) - 2 \,{\left (b^{2} c^{2} e x + b^{2} c^{2} d\right )} \log \left (F\right )^{2}\right )} \sqrt{e x + d} F^{b c x + a c}}{4 \, b^{3} c^{3} \log \left (F\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19088, size = 408, normalized size = 2.96 \begin{align*} \frac{1}{4} \,{\left (2 \, d{\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 )} - \frac{\sqrt{\pi }{\left (2 \, b c d e \log \left (F\right ) + 3 \, e^{2}\right )} \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 )} + 1\right )}}{\sqrt{-b c e \log \left (F\right )} b^{2} c^{2} \log \left (F\right )^{2}} + \frac{2 \,{\left (2 \,{\left (x e + d\right )}^{\frac{3}{2}} b c e \log \left (F\right ) - 2 \, \sqrt{x e + d} b c d e \log \left (F\right ) - 3 \, \sqrt{x e + d} e^{2}\right )} 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 )}\right )}}{b^{2} c^{2} \log \left (F\right )^{2}}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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