Integrand size = 13, antiderivative size = 44 \[ \int F^{a+b (c+d x)^2} \, dx=\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]
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Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2235} \[ \int F^{a+b (c+d x)^2} \, dx=\frac {\sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]
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Rule 2235
Rubi steps \begin{align*} \text {integral}& = \frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int F^{a+b (c+d x)^2} \, dx=\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{2 \sqrt {b} d \sqrt {\log (F)}} \]
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Time = 0.00 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.32
method | result | size |
risch | \(-\frac {\sqrt {\pi }\, F^{b \,c^{2}+a} F^{-b \,c^{2}} \operatorname {erf}\left (-d \sqrt {-b \ln \left (F \right )}\, x +\frac {b c \ln \left (F \right )}{\sqrt {-b \ln \left (F \right )}}\right )}{2 d \sqrt {-b \ln \left (F \right )}}\) | \(58\) |
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09 \[ \int F^{a+b (c+d x)^2} \, dx=-\frac {\sqrt {\pi } \sqrt {-b d^{2} \log \left (F\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \left (F\right )} {\left (d x + c\right )}}{d}\right )}{2 \, b d^{2} \log \left (F\right )} \]
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\[ \int F^{a+b (c+d x)^2} \, dx=\int F^{a + b \left (c + d x\right )^{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.32 \[ \int F^{a+b (c+d x)^2} \, dx=\frac {\sqrt {\pi } F^{b c^{2} + a} \operatorname {erf}\left (\sqrt {-b \log \left (F\right )} d x - \frac {b c \log \left (F\right )}{\sqrt {-b \log \left (F\right )}}\right )}{2 \, \sqrt {-b \log \left (F\right )} F^{b c^{2}} d} \]
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Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int F^{a+b (c+d x)^2} \, dx=-\frac {\sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} d {\left (x + \frac {c}{d}\right )}\right )}{2 \, \sqrt {-b \log \left (F\right )} d} \]
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Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09 \[ \int F^{a+b (c+d x)^2} \, dx=-\frac {F^a\,\sqrt {\pi }\,\mathrm {erf}\left (\frac {1{}\mathrm {i}\,b\,x\,\ln \left (F\right )\,d^2+1{}\mathrm {i}\,b\,c\,\ln \left (F\right )\,d}{\sqrt {b\,d^2\,\ln \left (F\right )}}\right )\,1{}\mathrm {i}}{2\,\sqrt {b\,d^2\,\ln \left (F\right )}} \]
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