Optimal. Leaf size=130 \[ \frac{4 \sqrt{\pi } b^{3/2} c^{3/2} \log ^{\frac{3}{2}}(F) F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{3 e^{5/2}}-\frac{4 b c \log (F) F^{c (a+b x)}}{3 e^2 \sqrt{d+e x}}-\frac{2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}} \]
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Rubi [A] time = 0.119372, antiderivative size = 130, 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 = {2177, 2180, 2204} \[ \frac{4 \sqrt{\pi } b^{3/2} c^{3/2} \log ^{\frac{3}{2}}(F) F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{3 e^{5/2}}-\frac{4 b c \log (F) F^{c (a+b x)}}{3 e^2 \sqrt{d+e x}}-\frac{2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}} \]
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)^{5/2}} \, dx &=-\frac{2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}}+\frac{(2 b c \log (F)) \int \frac{F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx}{3 e}\\ &=-\frac{2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}}-\frac{4 b c F^{c (a+b x)} \log (F)}{3 e^2 \sqrt{d+e x}}+\frac{\left (4 b^2 c^2 \log ^2(F)\right ) \int \frac{F^{c (a+b x)}}{\sqrt{d+e x}} \, dx}{3 e^2}\\ &=-\frac{2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}}-\frac{4 b c F^{c (a+b x)} \log (F)}{3 e^2 \sqrt{d+e x}}+\frac{\left (8 b^2 c^2 \log ^2(F)\right ) \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 )}{3 e^3}\\ &=-\frac{2 F^{c (a+b x)}}{3 e (d+e x)^{3/2}}-\frac{4 b c F^{c (a+b x)} \log (F)}{3 e^2 \sqrt{d+e x}}+\frac{4 b^{3/2} c^{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 ) \log ^{\frac{3}{2}}(F)}{3 e^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.234234, size = 92, normalized size = 0.71 \[ -\frac{2 \left (2 e F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{b c \log (F) (d+e x)}{e}\right )+F^{c (a+b x)} (2 b c \log (F) (d+e x)+e)\right )}{3 e^2 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int{{F}^{c \left ( bx+a \right ) } \left ( ex+d \right ) ^{-{\frac{5}{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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57267, size = 327, normalized size = 2.52 \begin{align*} -\frac{2 \,{\left (\frac{2 \, \sqrt{\pi }{\left (b c e^{2} x^{2} + 2 \, b c d e x + b c d^{2}\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 ) \log \left (F\right )}{F^{\frac{b c d - a c e}{e}}} + \sqrt{e x + d}{\left (2 \,{\left (b c e x + b c d\right )} \log \left (F\right ) + e\right )} F^{b c x + a c}\right )}}{3 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \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 [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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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