Optimal. Leaf size=49 \[ -\frac{F^a \left (-b \log (F) (c+d x)^2\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-b \log (F) (c+d x)^2\right )}{2 d (c+d x)^9} \]
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Rubi [A] time = 0.0613516, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2218} \[ -\frac{F^a \left (-b \log (F) (c+d x)^2\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-b \log (F) (c+d x)^2\right )}{2 d (c+d x)^9} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int \frac{F^{a+b (c+d x)^2}}{(c+d x)^{10}} \, dx &=-\frac{F^a \Gamma \left (-\frac{9}{2},-b (c+d x)^2 \log (F)\right ) \left (-b (c+d x)^2 \log (F)\right )^{9/2}}{2 d (c+d x)^9}\\ \end{align*}
Mathematica [A] time = 0.0299586, size = 49, normalized size = 1. \[ -\frac{F^a \left (-b \log (F) (c+d x)^2\right )^{9/2} \text{Gamma}\left (-\frac{9}{2},-b \log (F) (c+d x)^2\right )}{2 d (c+d x)^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.122, size = 195, normalized size = 4. \begin{align*} -{\frac{{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{9\,d \left ( dx+c \right ) ^{9}}}-{\frac{2\,b\ln \left ( F \right ){F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{63\,d \left ( dx+c \right ) ^{7}}}-{\frac{4\,{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{315\,d \left ( dx+c \right ) ^{5}}}-{\frac{8\,{b}^{3} \left ( \ln \left ( F \right ) \right ) ^{3}{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{945\,d \left ( dx+c \right ) ^{3}}}-{\frac{16\,{b}^{4} \left ( \ln \left ( F \right ) \right ) ^{4}{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{945\, \left ( dx+c \right ) d}}+{\frac{16\,{b}^{5} \left ( \ln \left ( F \right ) \right ) ^{5}\sqrt{\pi }{F}^{a}}{945\,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 )}^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.77045, size = 1273, normalized size = 25.98 \begin{align*} -\frac{16 \, \sqrt{\pi }{\left (b^{4} d^{9} x^{9} + 9 \, b^{4} c d^{8} x^{8} + 36 \, b^{4} c^{2} d^{7} x^{7} + 84 \, b^{4} c^{3} d^{6} x^{6} + 126 \, b^{4} c^{4} d^{5} x^{5} + 126 \, b^{4} c^{5} d^{4} x^{4} + 84 \, b^{4} c^{6} d^{3} x^{3} + 36 \, b^{4} c^{7} d^{2} x^{2} + 9 \, b^{4} c^{8} d x + b^{4} c^{9}\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 )^{4} +{\left (16 \,{\left (b^{4} d^{9} x^{8} + 8 \, b^{4} c d^{8} x^{7} + 28 \, b^{4} c^{2} d^{7} x^{6} + 56 \, b^{4} c^{3} d^{6} x^{5} + 70 \, b^{4} c^{4} d^{5} x^{4} + 56 \, b^{4} c^{5} d^{4} x^{3} + 28 \, b^{4} c^{6} d^{3} x^{2} + 8 \, b^{4} c^{7} d^{2} x + b^{4} c^{8} d\right )} \log \left (F\right )^{4} + 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} + 12 \,{\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} + 30 \,{\left (b d^{3} x^{2} + 2 \, b c d^{2} x + b c^{2} d\right )} \log \left (F\right ) + 105 \, d\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{945 \,{\left (d^{11} x^{9} + 9 \, c d^{10} x^{8} + 36 \, c^{2} d^{9} x^{7} + 84 \, c^{3} d^{8} x^{6} + 126 \, c^{4} d^{7} x^{5} + 126 \, c^{5} d^{6} x^{4} + 84 \, c^{6} d^{5} x^{3} + 36 \, c^{7} d^{4} x^{2} + 9 \, c^{8} d^{3} x + c^{9} 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 )}^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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