Integrand size = 12, antiderivative size = 56 \[ \int f^{a+b x+c x^2} \, dx=\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{2 \sqrt {c} \sqrt {\log (f)}} \]
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Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2266, 2235} \[ \int f^{a+b x+c x^2} \, dx=\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{2 \sqrt {c} \sqrt {\log (f)}} \]
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Rule 2235
Rule 2266
Rubi steps \begin{align*} \text {integral}& = f^{a-\frac {b^2}{4 c}} \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx \\ & = \frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{2 \sqrt {c} \sqrt {\log (f)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int f^{a+b x+c x^2} \, dx=\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{2 \sqrt {c} \sqrt {\log (f)}} \]
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Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{2 \sqrt {-c \ln \left (f \right )}}\) | \(50\) |
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Time = 0.34 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.98 \[ \int f^{a+b x+c x^2} \, dx=-\frac {\sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )}}{2 \, c}\right )}{2 \, c f^{\frac {b^{2} - 4 \, a c}{4 \, c}} \log \left (f\right )} \]
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\[ \int f^{a+b x+c x^2} \, dx=\int f^{a + b x + c x^{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89 \[ \int f^{a+b x+c x^2} \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right )}}\right )}{2 \, \sqrt {-c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} \]
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Time = 0.38 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.89 \[ \int f^{a+b x+c x^2} \, dx=-\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{2 \, \sqrt {-c \log \left (f\right )}} \]
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Time = 0.18 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88 \[ \int f^{a+b x+c x^2} \, dx=-\frac {f^a\,\sqrt {\pi }\,{\mathrm {e}}^{-\frac {b^2\,\ln \left (f\right )}{4\,c}}\,\mathrm {erf}\left (\frac {b\,\ln \left (f\right )\,1{}\mathrm {i}+c\,x\,\ln \left (f\right )\,2{}\mathrm {i}}{2\,\sqrt {c\,\ln \left (f\right )}}\right )\,1{}\mathrm {i}}{2\,\sqrt {c\,\ln \left (f\right )}} \]
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