Optimal. Leaf size=165 \[ \frac{8 \sqrt{\pi } b^{5/2} c^{5/2} \log ^{\frac{5}{2}}(F) F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{15 e^{7/2}}-\frac{8 b^2 c^2 \log ^2(F) F^{c (a+b x)}}{15 e^3 \sqrt{d+e x}}-\frac{4 b c \log (F) F^{c (a+b x)}}{15 e^2 (d+e x)^{3/2}}-\frac{2 F^{c (a+b x)}}{5 e (d+e x)^{5/2}} \]
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Rubi [A] time = 0.164663, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2177, 2180, 2204} \[ \frac{8 \sqrt{\pi } b^{5/2} c^{5/2} \log ^{\frac{5}{2}}(F) F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{15 e^{7/2}}-\frac{8 b^2 c^2 \log ^2(F) F^{c (a+b x)}}{15 e^3 \sqrt{d+e x}}-\frac{4 b c \log (F) F^{c (a+b x)}}{15 e^2 (d+e x)^{3/2}}-\frac{2 F^{c (a+b x)}}{5 e (d+e x)^{5/2}} \]
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)^{7/2}} \, dx &=-\frac{2 F^{c (a+b x)}}{5 e (d+e x)^{5/2}}+\frac{(2 b c \log (F)) \int \frac{F^{c (a+b x)}}{(d+e x)^{5/2}} \, dx}{5 e}\\ &=-\frac{2 F^{c (a+b x)}}{5 e (d+e x)^{5/2}}-\frac{4 b c F^{c (a+b x)} \log (F)}{15 e^2 (d+e x)^{3/2}}+\frac{\left (4 b^2 c^2 \log ^2(F)\right ) \int \frac{F^{c (a+b x)}}{(d+e x)^{3/2}} \, dx}{15 e^2}\\ &=-\frac{2 F^{c (a+b x)}}{5 e (d+e x)^{5/2}}-\frac{4 b c F^{c (a+b x)} \log (F)}{15 e^2 (d+e x)^{3/2}}-\frac{8 b^2 c^2 F^{c (a+b x)} \log ^2(F)}{15 e^3 \sqrt{d+e x}}+\frac{\left (8 b^3 c^3 \log ^3(F)\right ) \int \frac{F^{c (a+b x)}}{\sqrt{d+e x}} \, dx}{15 e^3}\\ &=-\frac{2 F^{c (a+b x)}}{5 e (d+e x)^{5/2}}-\frac{4 b c F^{c (a+b x)} \log (F)}{15 e^2 (d+e x)^{3/2}}-\frac{8 b^2 c^2 F^{c (a+b x)} \log ^2(F)}{15 e^3 \sqrt{d+e x}}+\frac{\left (16 b^3 c^3 \log ^3(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 )}{15 e^4}\\ &=-\frac{2 F^{c (a+b x)}}{5 e (d+e x)^{5/2}}-\frac{4 b c F^{c (a+b x)} \log (F)}{15 e^2 (d+e x)^{3/2}}-\frac{8 b^2 c^2 F^{c (a+b x)} \log ^2(F)}{15 e^3 \sqrt{d+e x}}+\frac{8 b^{5/2} c^{5/2} 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{5}{2}}(F)}{15 e^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.137881, size = 118, normalized size = 0.72 \[ \frac{2 \left (-2 b c \log (F) (d+e x) \left (2 e F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{b c \log (F) (d+e x)}{e}\right )+F^{c (a+b x)} (2 b c \log (F) (d+e x)+e)\right )-3 e^2 F^{c (a+b x)}\right )}{15 e^3 (d+e x)^{5/2}} \]
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 ) } \left ( ex+d \right ) ^{-{\frac{7}{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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56885, size = 497, normalized size = 3.01 \begin{align*} -\frac{2 \,{\left (\frac{4 \, \sqrt{\pi }{\left (b^{2} c^{2} e^{3} x^{3} + 3 \, b^{2} c^{2} d e^{2} x^{2} + 3 \, b^{2} c^{2} d^{2} e x + b^{2} c^{2} d^{3}\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 )^{2}}{F^{\frac{b c d - a c e}{e}}} +{\left (4 \,{\left (b^{2} c^{2} e^{2} x^{2} + 2 \, b^{2} c^{2} d e x + b^{2} c^{2} d^{2}\right )} \log \left (F\right )^{2} + 3 \, e^{2} + 2 \,{\left (b c e^{2} x + b c d e\right )} \log \left (F\right )\right )} \sqrt{e x + d} F^{b c x + a c}\right )}}{15 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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