Integrand size = 19, antiderivative size = 105 \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=-\frac {\sqrt {e} F^{c \left (a-\frac {b d}{e}\right )} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {d+e x} \sqrt {\log (F)}}{\sqrt {e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)}+\frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)} \]
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Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2207, 2211, 2235} \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\frac {\sqrt {d+e x} F^{c (a+b x)}}{b c \log (F)}-\frac {\sqrt {\pi } \sqrt {e} F^{c \left (a-\frac {b d}{e}\right )} \text {erfi}\left (\frac {\sqrt {b} \sqrt {c} \sqrt {\log (F)} \sqrt {d+e x}}{\sqrt {e}}\right )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)} \]
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Rule 2207
Rule 2211
Rule 2235
Rubi steps \begin{align*} \text {integral}& = \frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)}-\frac {e \int \frac {F^{c (a+b x)}}{\sqrt {d+e x}} \, dx}{2 b c \log (F)} \\ & = \frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)}-\frac {\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 )}{b c \log (F)} \\ & = -\frac {\sqrt {e} 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 )}{2 b^{3/2} c^{3/2} \log ^{\frac {3}{2}}(F)}+\frac {F^{c (a+b x)} \sqrt {d+e x}}{b c \log (F)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.60 \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=-\frac {F^{c \left (a-\frac {b d}{e}\right )} (d+e x)^{3/2} \Gamma \left (\frac {3}{2},-\frac {b c (d+e x) \log (F)}{e}\right )}{e \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{3/2}} \]
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\[\int F^{c \left (b x +a \right )} \sqrt {e x +d}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86 \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\frac {2 \, \sqrt {e x + d} F^{b c x + a c} b c \log \left (F\right ) + \frac {\sqrt {\pi } \sqrt {-\frac {b c \log \left (F\right )}{e}} e \operatorname {erf}\left (\sqrt {e x + d} \sqrt {-\frac {b c \log \left (F\right )}{e}}\right )}{F^{\frac {b c d - a c e}{e}}}}{2 \, b^{2} c^{2} \log \left (F\right )^{2}} \]
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\[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\int F^{c \left (a + b x\right )} \sqrt {d + e x}\, dx \]
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\[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} F^{{\left (b x + a\right )} c} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (81) = 162\).
Time = 0.32 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.80 \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=-\frac {\frac {2 \, \sqrt {\pi } d e \operatorname {erf}\left (-\frac {\sqrt {-b c e \log \left (F\right )} \sqrt {e x + d}}{e}\right ) e^{\left (-\frac {b c d \log \left (F\right ) - a c e \log \left (F\right )}{e}\right )}}{\sqrt {-b c e \log \left (F\right )}} - \frac {\sqrt {\pi } {\left (2 \, b c d \log \left (F\right ) + e\right )} e \operatorname {erf}\left (-\frac {\sqrt {-b c e \log \left (F\right )} \sqrt {e x + d}}{e}\right ) e^{\left (-\frac {b c d \log \left (F\right ) - a c e \log \left (F\right )}{e}\right )}}{\sqrt {-b c e \log \left (F\right )} b c \log \left (F\right )} - \frac {2 \, \sqrt {e x + d} e e^{\left (\frac {{\left (e x + d\right )} b c \log \left (F\right ) - b c d \log \left (F\right ) + a c e \log \left (F\right )}{e}\right )}}{b c \log \left (F\right )}}{2 \, e} \]
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Timed out. \[ \int F^{c (a+b x)} \sqrt {d+e x} \, dx=\int F^{c\,\left (a+b\,x\right )}\,\sqrt {d+e\,x} \,d x \]
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