Integrand size = 11, antiderivative size = 41 \[ \int f^{c (a+b x)^2} \, dx=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \]
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Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2235} \[ \int f^{c (a+b x)^2} \, dx=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \]
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Rule 2235
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int f^{c (a+b x)^2} \, dx=\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {c} (a+b x) \sqrt {\log (f)}\right )}{2 b \sqrt {c} \sqrt {\log (f)}} \]
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Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {\sqrt {\pi }\, \operatorname {erf}\left (-b \sqrt {-c \ln \left (f \right )}\, x +\frac {a c \ln \left (f \right )}{\sqrt {-c \ln \left (f \right )}}\right )}{2 b \sqrt {-c \ln \left (f \right )}}\) | \(41\) |
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none
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.10 \[ \int f^{c (a+b x)^2} \, dx=-\frac {\sqrt {\pi } \sqrt {-b^{2} c \log \left (f\right )} \operatorname {erf}\left (\frac {\sqrt {-b^{2} c \log \left (f\right )} {\left (b x + a\right )}}{b}\right )}{2 \, b^{2} c \log \left (f\right )} \]
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\[ \int f^{c (a+b x)^2} \, dx=\int f^{c \left (a + b x\right )^{2}}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.98 \[ \int f^{c (a+b x)^2} \, dx=\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} b x - \frac {a c \log \left (f\right )}{\sqrt {-c \log \left (f\right )}}\right )}{2 \, \sqrt {-c \log \left (f\right )} b} \]
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none
Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80 \[ \int f^{c (a+b x)^2} \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} b {\left (x + \frac {a}{b}\right )}\right )}{2 \, \sqrt {-c \log \left (f\right )} b} \]
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Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.10 \[ \int f^{c (a+b x)^2} \, dx=-\frac {\sqrt {\pi }\,\mathrm {erf}\left (\frac {1{}\mathrm {i}\,c\,x\,\ln \left (f\right )\,b^2+1{}\mathrm {i}\,a\,c\,\ln \left (f\right )\,b}{\sqrt {b^2\,c\,\ln \left (f\right )}}\right )\,1{}\mathrm {i}}{2\,\sqrt {b^2\,c\,\ln \left (f\right )}} \]
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