Optimal. Leaf size=170 \[ \frac{8 \sqrt{\pi } b^{7/2} F^a \log ^{\frac{7}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{105 d}-\frac{8 b^3 \log ^3(F) F^{a+b (c+d x)^2}}{105 d (c+d x)}-\frac{4 b^2 \log ^2(F) F^{a+b (c+d x)^2}}{105 d (c+d x)^3}-\frac{F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac{2 b \log (F) F^{a+b (c+d x)^2}}{35 d (c+d x)^5} \]
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Rubi [A] time = 0.285971, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2214, 2204} \[ \frac{8 \sqrt{\pi } b^{7/2} F^a \log ^{\frac{7}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{105 d}-\frac{8 b^3 \log ^3(F) F^{a+b (c+d x)^2}}{105 d (c+d x)}-\frac{4 b^2 \log ^2(F) F^{a+b (c+d x)^2}}{105 d (c+d x)^3}-\frac{F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac{2 b \log (F) F^{a+b (c+d x)^2}}{35 d (c+d x)^5} \]
Antiderivative was successfully verified.
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Rule 2214
Rule 2204
Rubi steps
\begin{align*} \int \frac{F^{a+b (c+d x)^2}}{(c+d x)^8} \, dx &=-\frac{F^{a+b (c+d x)^2}}{7 d (c+d x)^7}+\frac{1}{7} (2 b \log (F)) \int \frac{F^{a+b (c+d x)^2}}{(c+d x)^6} \, dx\\ &=-\frac{F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac{2 b F^{a+b (c+d x)^2} \log (F)}{35 d (c+d x)^5}+\frac{1}{35} \left (4 b^2 \log ^2(F)\right ) \int \frac{F^{a+b (c+d x)^2}}{(c+d x)^4} \, dx\\ &=-\frac{F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac{2 b F^{a+b (c+d x)^2} \log (F)}{35 d (c+d x)^5}-\frac{4 b^2 F^{a+b (c+d x)^2} \log ^2(F)}{105 d (c+d x)^3}+\frac{1}{105} \left (8 b^3 \log ^3(F)\right ) \int \frac{F^{a+b (c+d x)^2}}{(c+d x)^2} \, dx\\ &=-\frac{F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac{2 b F^{a+b (c+d x)^2} \log (F)}{35 d (c+d x)^5}-\frac{4 b^2 F^{a+b (c+d x)^2} \log ^2(F)}{105 d (c+d x)^3}-\frac{8 b^3 F^{a+b (c+d x)^2} \log ^3(F)}{105 d (c+d x)}+\frac{1}{105} \left (16 b^4 \log ^4(F)\right ) \int F^{a+b (c+d x)^2} \, dx\\ &=-\frac{F^{a+b (c+d x)^2}}{7 d (c+d x)^7}-\frac{2 b F^{a+b (c+d x)^2} \log (F)}{35 d (c+d x)^5}-\frac{4 b^2 F^{a+b (c+d x)^2} \log ^2(F)}{105 d (c+d x)^3}-\frac{8 b^3 F^{a+b (c+d x)^2} \log ^3(F)}{105 d (c+d x)}+\frac{8 b^{7/2} F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} (c+d x) \sqrt{\log (F)}\right ) \log ^{\frac{7}{2}}(F)}{105 d}\\ \end{align*}
Mathematica [A] time = 0.152442, size = 112, normalized size = 0.66 \[ \frac{F^a \left (8 \sqrt{\pi } b^{7/2} \log ^{\frac{7}{2}}(F) \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )+\frac{F^{b (c+d x)^2} \left (-8 b^3 \log ^3(F) (c+d x)^6-4 b^2 \log ^2(F) (c+d x)^4-6 b \log (F) (c+d x)^2-15\right )}{(c+d x)^7}\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.09, size = 162, normalized size = 1. \begin{align*} -{\frac{{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{7\,d \left ( dx+c \right ) ^{7}}}-{\frac{2\,b\ln \left ( F \right ){F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{35\,d \left ( dx+c \right ) ^{5}}}-{\frac{4\,{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{105\,d \left ( dx+c \right ) ^{3}}}-{\frac{8\,{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{105\, \left ( dx+c \right ) d}}+{\frac{8\,{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}\sqrt{\pi }{F}^{a}}{105\,d}{\it Erf} \left ( \sqrt{-b\ln \left ( F \right ) } \left ( dx+c \right ) \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \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 (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.60913, size = 913, normalized size = 5.37 \begin{align*} -\frac{8 \, \sqrt{\pi }{\left (b^{3} d^{7} x^{7} + 7 \, b^{3} c d^{6} x^{6} + 21 \, b^{3} c^{2} d^{5} x^{5} + 35 \, b^{3} c^{3} d^{4} x^{4} + 35 \, b^{3} c^{4} d^{3} x^{3} + 21 \, b^{3} c^{5} d^{2} x^{2} + 7 \, b^{3} c^{6} d x + b^{3} c^{7}\right )} \sqrt{-b d^{2} \log \left (F\right )} F^{a} \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right ) \log \left (F\right )^{3} +{\left (8 \,{\left (b^{3} d^{7} x^{6} + 6 \, b^{3} c d^{6} x^{5} + 15 \, b^{3} c^{2} d^{5} x^{4} + 20 \, b^{3} c^{3} d^{4} x^{3} + 15 \, b^{3} c^{4} d^{3} x^{2} + 6 \, b^{3} c^{5} d^{2} x + b^{3} c^{6} d\right )} \log \left (F\right )^{3} + 4 \,{\left (b^{2} d^{5} x^{4} + 4 \, b^{2} c d^{4} x^{3} + 6 \, b^{2} c^{2} d^{3} x^{2} + 4 \, b^{2} c^{3} d^{2} x + b^{2} c^{4} d\right )} \log \left (F\right )^{2} + 6 \,{\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} \log \left (F\right ) + 15 \, d\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{105 \,{\left (d^{9} x^{7} + 7 \, c d^{8} x^{6} + 21 \, c^{2} d^{7} x^{5} + 35 \, c^{3} d^{6} x^{4} + 35 \, c^{4} d^{5} x^{3} + 21 \, c^{5} d^{4} x^{2} + 7 \, c^{6} d^{3} x + c^{7} d^{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 (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{8}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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