Integrand size = 19, antiderivative size = 165 \[ \int \frac {F^{c (a+b x)}}{(d+e x)^{7/2}} \, dx=-\frac {2 F^{c (a+b x)}}{5 e (d+e x)^{5/2}}-\frac {4 b c F^{c (a+b x)} \log (F)}{15 e^2 (d+e x)^{3/2}}-\frac {8 b^2 c^2 F^{c (a+b x)} \log ^2(F)}{15 e^3 \sqrt {d+e x}}+\frac {8 b^{5/2} c^{5/2} F^{c \left (a-\frac {b d}{e}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {d+e x} \sqrt {\log (F)}}{\sqrt {e}}\right ) \log ^{\frac {5}{2}}(F)}{15 e^{7/2}} \]
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Time = 0.11 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2208, 2211, 2235} \[ \int \frac {F^{c (a+b x)}}{(d+e x)^{7/2}} \, dx=\frac {8 \sqrt {\pi } b^{5/2} c^{5/2} \log ^{\frac {5}{2}}(F) F^{c \left (a-\frac {b d}{e}\right )} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{15 e^{7/2}}-\frac {8 b^2 c^2 \log ^2(F) F^{c (a+b x)}}{15 e^3 \sqrt {d+e x}}-\frac {4 b c \log (F) F^{c (a+b x)}}{15 e^2 (d+e x)^{3/2}}-\frac {2 F^{c (a+b x)}}{5 e (d+e x)^{5/2}} \]
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Rule 2208
Rule 2211
Rule 2235
Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {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*}
Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.72 \[ \int \frac {F^{c (a+b x)}}{(d+e x)^{7/2}} \, dx=\frac {2 \left (-3 e^2 F^{c (a+b x)}-2 b c (d+e x) \log (F) \left (2 e F^{c \left (a-\frac {b d}{e}\right )} \Gamma \left (\frac {1}{2},-\frac {b c (d+e x) \log (F)}{e}\right ) \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{3/2}+F^{c (a+b x)} (e+2 b c (d+e x) \log (F))\right )\right )}{15 e^3 (d+e x)^{5/2}} \]
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\[\int \frac {F^{c \left (b x +a \right )}}{\left (e x +d \right )^{\frac {7}{2}}}d x\]
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Time = 0.26 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.39 \[ \int \frac {F^{c (a+b x)}}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (\frac {4 \, \sqrt {\pi } {\left (b^{2} c^{2} e^{3} x^{3} + 3 \, b^{2} c^{2} d e^{2} x^{2} + 3 \, b^{2} c^{2} d^{2} e x + b^{2} c^{2} d^{3}\right )} \sqrt {-\frac {b c \log \left (F\right )}{e}} \operatorname {erf}\left (\sqrt {e x + d} \sqrt {-\frac {b c \log \left (F\right )}{e}}\right ) \log \left (F\right )^{2}}{F^{\frac {b c d - a c e}{e}}} + {\left (4 \, {\left (b^{2} c^{2} e^{2} x^{2} + 2 \, b^{2} c^{2} d e x + b^{2} c^{2} d^{2}\right )} \log \left (F\right )^{2} + 3 \, e^{2} + 2 \, {\left (b c e^{2} x + b c d e\right )} \log \left (F\right )\right )} \sqrt {e x + d} F^{b c x + a c}\right )}}{15 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
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\[ \int \frac {F^{c (a+b x)}}{(d+e x)^{7/2}} \, dx=\int \frac {F^{c \left (a + b x\right )}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {F^{c (a+b x)}}{(d+e x)^{7/2}} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {F^{c (a+b x)}}{(d+e x)^{7/2}} \, dx=\int { \frac {F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {F^{c (a+b x)}}{(d+e x)^{7/2}} \, dx=\int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]
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