Integrand size = 20, antiderivative size = 75 \[ \int f^{b x+c x^2} (b+2 c x)^2 \, dx=-\frac {\sqrt {c} f^{-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{\log ^{\frac {3}{2}}(f)}+\frac {f^{b x+c x^2} (b+2 c x)}{\log (f)} \]
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Time = 0.04 (sec) , antiderivative size = 75, 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 = {2269, 2266, 2235} \[ \int f^{b x+c x^2} (b+2 c x)^2 \, dx=\frac {(b+2 c x) f^{b x+c x^2}}{\log (f)}-\frac {\sqrt {\pi } \sqrt {c} f^{-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{\log ^{\frac {3}{2}}(f)} \]
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Rule 2235
Rule 2266
Rule 2269
Rubi steps \begin{align*} \text {integral}& = \frac {f^{b x+c x^2} (b+2 c x)}{\log (f)}-\frac {(2 c) \int f^{b x+c x^2} \, dx}{\log (f)} \\ & = \frac {f^{b x+c x^2} (b+2 c x)}{\log (f)}-\frac {\left (2 c f^{-\frac {b^2}{4 c}}\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{\log (f)} \\ & = -\frac {\sqrt {c} f^{-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{\log ^{\frac {3}{2}}(f)}+\frac {f^{b x+c x^2} (b+2 c x)}{\log (f)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.12 \[ \int f^{b x+c x^2} (b+2 c x)^2 \, dx=\frac {f^{-\frac {b^2}{4 c}} \left (-\sqrt {c} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )+f^{\frac {(b+2 c x)^2}{4 c}} (b+2 c x) \sqrt {\log (f)}\right )}{\log ^{\frac {3}{2}}(f)} \]
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Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.20
method | result | size |
risch | \(\frac {2 c x \,f^{b x} f^{c \,x^{2}}}{\ln \left (f \right )}+\frac {b \,f^{b x} f^{c \,x^{2}}}{\ln \left (f \right )}+\frac {c \sqrt {\pi }\, f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{\ln \left (f \right ) \sqrt {-c \ln \left (f \right )}}\) | \(90\) |
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Time = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.91 \[ \int f^{b x+c x^2} (b+2 c x)^2 \, dx=\frac {{\left (2 \, c x + b\right )} f^{c x^{2} + b x} \log \left (f\right ) + \frac {\sqrt {\pi } \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac {b^{2}}{4 \, c}}}}{\log \left (f\right )^{2}} \]
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\[ \int f^{b x+c x^2} (b+2 c x)^2 \, dx=\int f^{b x + c x^{2}} \left (b + 2 c x\right )^{2}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (63) = 126\).
Time = 0.33 (sec) , antiderivative size = 329, normalized size of antiderivative = 4.39 \[ \int f^{b x+c x^2} (b+2 c x)^2 \, dx=\frac {\sqrt {\pi } b^{2} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right )}}\right )}{2 \, \sqrt {-c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{2}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac {3}{2}}} - \frac {2 \, c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )}{\left (c \log \left (f\right )\right )^{\frac {3}{2}}}\right )} b c}{\sqrt {c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} + \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{3}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{4 \, c}\right ) \log \left (f\right )^{3}}{\left (-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}\right )^{\frac {3}{2}} \left (c \log \left (f\right )\right )^{\frac {5}{2}}} - \frac {4 \, b c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )^{2}}{\left (c \log \left (f\right )\right )^{\frac {5}{2}}}\right )} c^{2}}{2 \, \sqrt {c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} \]
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Time = 0.32 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int f^{b x+c x^2} (b+2 c x)^2 \, dx=\frac {c {\left (2 \, x + \frac {b}{c}\right )} e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right )\right )}}{\log \left (f\right )} + \frac {\sqrt {\pi } c \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b}{c}\right )}\right )}{\sqrt {-c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}} \log \left (f\right )} \]
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Time = 0.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.15 \[ \int f^{b x+c x^2} (b+2 c x)^2 \, dx=\frac {b\,f^{c\,x^2}\,f^{b\,x}}{\ln \left (f\right )}+\frac {2\,c\,f^{c\,x^2}\,f^{b\,x}\,x}{\ln \left (f\right )}-\frac {c\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b\,\ln \left (f\right )}{2}+c\,x\,\ln \left (f\right )}{\sqrt {c\,\ln \left (f\right )}}\right )}{f^{\frac {b^2}{4\,c}}\,\ln \left (f\right )\,\sqrt {c\,\ln \left (f\right )}} \]
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