Integrand size = 20, antiderivative size = 81 \[ \int \frac {f^{b x+c x^2}}{(b+2 c x)^2} \, dx=-\frac {f^{b x+c x^2}}{2 c (b+2 c x)}+\frac {f^{-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \sqrt {\log (f)}}{4 c^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2271, 2266, 2235} \[ \int \frac {f^{b x+c x^2}}{(b+2 c x)^2} \, dx=\frac {\sqrt {\pi } \sqrt {\log (f)} f^{-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2}}-\frac {f^{b x+c x^2}}{2 c (b+2 c x)} \]
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Rule 2235
Rule 2266
Rule 2271
Rubi steps \begin{align*} \text {integral}& = -\frac {f^{b x+c x^2}}{2 c (b+2 c x)}+\frac {\log (f) \int f^{b x+c x^2} \, dx}{2 c} \\ & = -\frac {f^{b x+c x^2}}{2 c (b+2 c x)}+\frac {\left (f^{-\frac {b^2}{4 c}} \log (f)\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{2 c} \\ & = -\frac {f^{b x+c x^2}}{2 c (b+2 c x)}+\frac {f^{-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \sqrt {\log (f)}}{4 c^{3/2}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.16 \[ \int \frac {f^{b x+c x^2}}{(b+2 c x)^2} \, dx=\frac {f^{-\frac {b^2}{4 c}} \left (-2 \sqrt {c} f^{\frac {(b+2 c x)^2}{4 c}}+\sqrt {\pi } (b+2 c x) \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \sqrt {\log (f)}\right )}{4 c^{3/2} (b+2 c x)} \]
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Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.07
method | result | size |
risch | \(-\frac {f^{\frac {\left (2 x c +b \right )^{2}}{4 c}} f^{-\frac {b^{2}}{4 c}}}{2 c \left (2 x c +b \right )}+\frac {\ln \left (f \right ) \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (\frac {\sqrt {-\frac {\ln \left (f \right )}{c}}\, \left (2 x c +b \right )}{2}\right )}{4 c^{2} \sqrt {-\frac {\ln \left (f \right )}{c}}}\) | \(87\) |
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Time = 0.35 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98 \[ \int \frac {f^{b x+c x^2}}{(b+2 c x)^2} \, dx=-\frac {2 \, c f^{c x^{2} + b x} + \frac {\sqrt {\pi } {\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac {b^{2}}{4 \, c}}}}{4 \, {\left (2 \, c^{3} x + b c^{2}\right )}} \]
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\[ \int \frac {f^{b x+c x^2}}{(b+2 c x)^2} \, dx=\int \frac {f^{b x + c x^{2}}}{\left (b + 2 c x\right )^{2}}\, dx \]
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\[ \int \frac {f^{b x+c x^2}}{(b+2 c x)^2} \, dx=\int { \frac {f^{c x^{2} + b x}}{{\left (2 \, c x + b\right )}^{2}} \,d x } \]
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\[ \int \frac {f^{b x+c x^2}}{(b+2 c x)^2} \, dx=\int { \frac {f^{c x^{2} + b x}}{{\left (2 \, c x + b\right )}^{2}} \,d x } \]
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Time = 0.43 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90 \[ \int \frac {f^{b x+c x^2}}{(b+2 c x)^2} \, dx=\frac {\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b\,\ln \left (f\right )}{2}+c\,x\,\ln \left (f\right )}{\sqrt {c\,\ln \left (f\right )}}\right )\,\ln \left (f\right )}{4\,c\,f^{\frac {b^2}{4\,c}}\,\sqrt {c\,\ln \left (f\right )}}-\frac {f^{c\,x^2}\,f^{b\,x}}{2\,c\,\left (b+2\,c\,x\right )} \]
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