Optimal. Leaf size=208 \[ \frac{105 \sqrt{\pi } e^{7/2} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{16 b^{9/2} c^{9/2} \log ^{\frac{9}{2}}(F)}+\frac{35 e^2 (d+e x)^{3/2} F^{c (a+b x)}}{4 b^3 c^3 \log ^3(F)}-\frac{105 e^3 \sqrt{d+e x} F^{c (a+b x)}}{8 b^4 c^4 \log ^4(F)}-\frac{7 e (d+e x)^{5/2} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac{(d+e x)^{7/2} F^{c (a+b x)}}{b c \log (F)} \]
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Rubi [A] time = 0.266096, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2176, 2180, 2204} \[ \frac{105 \sqrt{\pi } e^{7/2} F^{c \left (a-\frac{b d}{e}\right )} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{c} \sqrt{\log (F)} \sqrt{d+e x}}{\sqrt{e}}\right )}{16 b^{9/2} c^{9/2} \log ^{\frac{9}{2}}(F)}+\frac{35 e^2 (d+e x)^{3/2} F^{c (a+b x)}}{4 b^3 c^3 \log ^3(F)}-\frac{105 e^3 \sqrt{d+e x} F^{c (a+b x)}}{8 b^4 c^4 \log ^4(F)}-\frac{7 e (d+e x)^{5/2} F^{c (a+b x)}}{2 b^2 c^2 \log ^2(F)}+\frac{(d+e x)^{7/2} F^{c (a+b x)}}{b c \log (F)} \]
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)^{7/2} \, dx &=\frac{F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)}-\frac{(7 e) \int F^{c (a+b x)} (d+e x)^{5/2} \, dx}{2 b c \log (F)}\\ &=-\frac{7 e F^{c (a+b x)} (d+e x)^{5/2}}{2 b^2 c^2 \log ^2(F)}+\frac{F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)}+\frac{\left (35 e^2\right ) \int F^{c (a+b x)} (d+e x)^{3/2} \, dx}{4 b^2 c^2 \log ^2(F)}\\ &=\frac{35 e^2 F^{c (a+b x)} (d+e x)^{3/2}}{4 b^3 c^3 \log ^3(F)}-\frac{7 e F^{c (a+b x)} (d+e x)^{5/2}}{2 b^2 c^2 \log ^2(F)}+\frac{F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)}-\frac{\left (105 e^3\right ) \int F^{c (a+b x)} \sqrt{d+e x} \, dx}{8 b^3 c^3 \log ^3(F)}\\ &=-\frac{105 e^3 F^{c (a+b x)} \sqrt{d+e x}}{8 b^4 c^4 \log ^4(F)}+\frac{35 e^2 F^{c (a+b x)} (d+e x)^{3/2}}{4 b^3 c^3 \log ^3(F)}-\frac{7 e F^{c (a+b x)} (d+e x)^{5/2}}{2 b^2 c^2 \log ^2(F)}+\frac{F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)}+\frac{\left (105 e^4\right ) \int \frac{F^{c (a+b x)}}{\sqrt{d+e x}} \, dx}{16 b^4 c^4 \log ^4(F)}\\ &=-\frac{105 e^3 F^{c (a+b x)} \sqrt{d+e x}}{8 b^4 c^4 \log ^4(F)}+\frac{35 e^2 F^{c (a+b x)} (d+e x)^{3/2}}{4 b^3 c^3 \log ^3(F)}-\frac{7 e F^{c (a+b x)} (d+e x)^{5/2}}{2 b^2 c^2 \log ^2(F)}+\frac{F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)}+\frac{\left (105 e^3\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 )}{8 b^4 c^4 \log ^4(F)}\\ &=\frac{105 e^{7/2} 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 )}{16 b^{9/2} c^{9/2} \log ^{\frac{9}{2}}(F)}-\frac{105 e^3 F^{c (a+b x)} \sqrt{d+e x}}{8 b^4 c^4 \log ^4(F)}+\frac{35 e^2 F^{c (a+b x)} (d+e x)^{3/2}}{4 b^3 c^3 \log ^3(F)}-\frac{7 e F^{c (a+b x)} (d+e x)^{5/2}}{2 b^2 c^2 \log ^2(F)}+\frac{F^{c (a+b x)} (d+e x)^{7/2}}{b c \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0296248, size = 72, normalized size = 0.35 \[ \frac{e^4 F^{c \left (a-\frac{b d}{e}\right )} \sqrt{-\frac{b c \log (F) (d+e x)}{e}} \text{Gamma}\left (\frac{9}{2},-\frac{b c \log (F) (d+e x)}{e}\right )}{b^5 c^5 \log ^5(F) \sqrt{d+e x}} \]
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{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{\left (e x + d\right )}^{\frac{7}{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 = 1.55591, size = 516, normalized size = 2.48 \begin{align*} -\frac{\frac{105 \, \sqrt{\pi } \sqrt{-\frac{b c \log \left (F\right )}{e}} e^{4} \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 (105 \, b c e^{3} \log \left (F\right ) - 8 \,{\left (b^{4} c^{4} e^{3} x^{3} + 3 \, b^{4} c^{4} d e^{2} x^{2} + 3 \, b^{4} c^{4} d^{2} e x + b^{4} c^{4} d^{3}\right )} \log \left (F\right )^{4} + 28 \,{\left (b^{3} c^{3} e^{3} x^{2} + 2 \, b^{3} c^{3} d e^{2} x + b^{3} c^{3} d^{2} e\right )} \log \left (F\right )^{3} - 70 \,{\left (b^{2} c^{2} e^{3} x + b^{2} c^{2} d e^{2}\right )} \log \left (F\right )^{2}\right )} \sqrt{e x + d} F^{b c x + a c}}{16 \, b^{5} c^{5} \log \left (F\right )^{5}} \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 = 1.37808, size = 1301, normalized size = 6.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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