Integrand size = 16, antiderivative size = 164 \[ \int f^{a+b x+c x^2} x^2 \, dx=-\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}-\frac {b f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac {f^{a+b x+c x^2} x}{2 c \log (f)}+\frac {b^2 f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{8 c^{5/2} \sqrt {\log (f)}} \]
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Time = 0.06 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2273, 2272, 2266, 2235} \[ \int f^{a+b x+c x^2} x^2 \, dx=-\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}+\frac {\sqrt {\pi } b^2 f^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{8 c^{5/2} \sqrt {\log (f)}}-\frac {b f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac {x f^{a+b x+c x^2}}{2 c \log (f)} \]
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Rule 2235
Rule 2266
Rule 2272
Rule 2273
Rubi steps \begin{align*} \text {integral}& = \frac {f^{a+b x+c x^2} x}{2 c \log (f)}-\frac {b \int f^{a+b x+c x^2} x \, dx}{2 c}-\frac {\int f^{a+b x+c x^2} \, dx}{2 c \log (f)} \\ & = -\frac {b f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac {f^{a+b x+c x^2} x}{2 c \log (f)}+\frac {b^2 \int f^{a+b x+c x^2} \, dx}{4 c^2}-\frac {f^{a-\frac {b^2}{4 c}} \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{2 c \log (f)} \\ & = -\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 )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}-\frac {b f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac {f^{a+b x+c x^2} x}{2 c \log (f)}+\frac {\left (b^2 f^{a-\frac {b^2}{4 c}}\right ) \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx}{4 c^2} \\ & = -\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 )}{4 c^{3/2} \log ^{\frac {3}{2}}(f)}-\frac {b f^{a+b x+c x^2}}{4 c^2 \log (f)}+\frac {f^{a+b x+c x^2} x}{2 c \log (f)}+\frac {b^2 f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{8 c^{5/2} \sqrt {\log (f)}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.63 \[ \int f^{a+b x+c x^2} x^2 \, dx=\frac {f^{a-\frac {b^2}{4 c}} \left (-2 \sqrt {c} f^{\frac {(b+2 c x)^2}{4 c}} (b-2 c x) \sqrt {\log (f)}+\sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right ) \left (-2 c+b^2 \log (f)\right )\right )}{8 c^{5/2} \log ^{\frac {3}{2}}(f)} \]
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Time = 0.04 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.99
method | result | size |
risch | \(\frac {x \,f^{c \,x^{2}} f^{b x} f^{a}}{2 c \ln \left (f \right )}-\frac {b \,f^{c \,x^{2}} f^{b x} f^{a}}{4 c^{2} \ln \left (f \right )}-\frac {b^{2} \sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{8 c^{2} \sqrt {-c \ln \left (f \right )}}+\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 c \ln \left (f \right ) \sqrt {-c \ln \left (f \right )}}\) | \(163\) |
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Time = 0.36 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.58 \[ \int f^{a+b x+c x^2} x^2 \, dx=\frac {2 \, {\left (2 \, c^{2} x - b c\right )} f^{c x^{2} + b x + a} \log \left (f\right ) - \frac {\sqrt {\pi } {\left (b^{2} \log \left (f\right ) - 2 \, c\right )} \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )}}{2 \, c}\right )}{f^{\frac {b^{2} - 4 \, a c}{4 \, c}}}}{8 \, c^{3} \log \left (f\right )^{2}} \]
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\[ \int f^{a+b x+c x^2} x^2 \, dx=\int f^{a + b x + c x^{2}} x^{2}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.01 \[ \int f^{a+b x+c x^2} x^2 \, dx=\frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}}\right ) - 1\right )} \log \left (f\right )^{3}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}} \left (c \log \left (f\right )\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{4 \, c}\right ) \log \left (f\right )^{3}}{\left (-\frac {{\left (2 \, c x + b\right )}^{2} \log \left (f\right )}{c}\right )^{\frac {3}{2}} \left (c \log \left (f\right )\right )^{\frac {5}{2}}} - \frac {4 \, b c f^{\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}} \log \left (f\right )^{2}}{\left (c \log \left (f\right )\right )^{\frac {5}{2}}}\right )} f^{a - \frac {b^{2}}{4 \, c}}}{8 \, \sqrt {c \log \left (f\right )}} \]
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Time = 0.35 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.66 \[ \int f^{a+b x+c x^2} x^2 \, dx=-\frac {\frac {\sqrt {\pi } {\left (b^{2} \log \left (f\right ) - 2 \, c\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{\sqrt {-c \log \left (f\right )} \log \left (f\right )} - \frac {2 \, {\left (c {\left (2 \, x + \frac {b}{c}\right )} - 2 \, b\right )} e^{\left (c x^{2} \log \left (f\right ) + b x \log \left (f\right ) + a \log \left (f\right )\right )}}{\log \left (f\right )}}{8 \, c^{2}} \]
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Time = 0.41 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.68 \[ \int f^{a+b x+c x^2} x^2 \, dx=\frac {f^a\,f^{c\,x^2}\,f^{b\,x}\,x}{2\,c\,\ln \left (f\right )}-\frac {b\,f^a\,f^{c\,x^2}\,f^{b\,x}}{4\,c^2\,\ln \left (f\right )}-\frac {f^{a-\frac {b^2}{4\,c}}\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {\frac {b\,\ln \left (f\right )}{2}+c\,x\,\ln \left (f\right )}{\sqrt {c\,\ln \left (f\right )}}\right )\,\left (\frac {c}{4}-\frac {b^2\,\ln \left (f\right )}{8}\right )}{c^2\,\ln \left (f\right )\,\sqrt {c\,\ln \left (f\right )}} \]
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