Integrand size = 21, antiderivative size = 77 \[ \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx=-\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{4 b^{3/2} d \log ^{\frac {3}{2}}(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)}{2 b d \log (F)} \]
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Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2243, 2235} \[ \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx=\frac {(c+d x) F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {\sqrt {\pi } F^a \text {erfi}\left (\sqrt {b} \sqrt {\log (F)} (c+d x)\right )}{4 b^{3/2} d \log ^{\frac {3}{2}}(F)} \]
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Rule 2235
Rule 2243
Rubi steps \begin{align*} \text {integral}& = \frac {F^{a+b (c+d x)^2} (c+d x)}{2 b d \log (F)}-\frac {\int F^{a+b (c+d x)^2} \, dx}{2 b \log (F)} \\ & = -\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{4 b^{3/2} d \log ^{\frac {3}{2}}(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)}{2 b d \log (F)} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00 \[ \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx=-\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} (c+d x) \sqrt {\log (F)}\right )}{4 b^{3/2} d \log ^{\frac {3}{2}}(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)}{2 b d \log (F)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(145\) vs. \(2(63)=126\).
Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.90
method | result | size |
risch | \(\frac {F^{b \,c^{2}} F^{a} x \,F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 \ln \left (F \right ) b}+\frac {F^{b \,c^{2}} F^{a} c \,F^{b \,d^{2} x^{2}} F^{2 b c d x}}{2 d \ln \left (F \right ) b}+\frac {F^{b \,c^{2}} F^{a} \sqrt {\pi }\, F^{-b \,c^{2}} \operatorname {erf}\left (-d \sqrt {-b \ln \left (F \right )}\, x +\frac {b c \ln \left (F \right )}{\sqrt {-b \ln \left (F \right )}}\right )}{4 d \ln \left (F \right ) b \sqrt {-b \ln \left (F \right )}}\) | \(146\) |
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Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.14 \[ \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx=\frac {\sqrt {\pi } \sqrt {-b d^{2} \log \left (F\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \left (F\right )} {\left (d x + c\right )}}{d}\right ) + 2 \, {\left (b d^{2} x + b c d\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a} \log \left (F\right )}{4 \, b^{2} d^{2} \log \left (F\right )^{2}} \]
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\[ \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx=\int F^{a + b \left (c + d x\right )^{2}} \left (c + d x\right )^{2}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (63) = 126\).
Time = 0.36 (sec) , antiderivative size = 413, normalized size of antiderivative = 5.36 \[ \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx=-\frac {{\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b c {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{2}}{\left (b \log \left (F\right )\right )^{\frac {3}{2}} d^{2} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}} - \frac {F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b \log \left (F\right )}{\left (b \log \left (F\right )\right )^{\frac {3}{2}} d}\right )} F^{a} c}{\sqrt {b \log \left (F\right )}} + \frac {{\left (\frac {\sqrt {\pi } {\left (b d^{2} x + b c d\right )} b^{2} c^{2} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}\right ) - 1\right )} \log \left (F\right )^{3}}{\left (b \log \left (F\right )\right )^{\frac {5}{2}} d^{3} \sqrt {-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}}} - \frac {2 \, F^{\frac {{\left (b d^{2} x + b c d\right )}^{2}}{b d^{2}}} b^{2} c \log \left (F\right )^{2}}{\left (b \log \left (F\right )\right )^{\frac {5}{2}} d^{2}} - \frac {{\left (b d^{2} x + b c d\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right ) \log \left (F\right )^{3}}{\left (b \log \left (F\right )\right )^{\frac {5}{2}} d^{5} \left (-\frac {{\left (b d^{2} x + b c d\right )}^{2} \log \left (F\right )}{b d^{2}}\right )^{\frac {3}{2}}}\right )} F^{a} d}{2 \, \sqrt {b \log \left (F\right )}} + \frac {\sqrt {\pi } F^{b c^{2} + a} c^{2} \operatorname {erf}\left (\sqrt {-b \log \left (F\right )} d x - \frac {b c \log \left (F\right )}{\sqrt {-b \log \left (F\right )}}\right )}{2 \, \sqrt {-b \log \left (F\right )} F^{b c^{2}} d} \]
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Time = 0.32 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.18 \[ \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx=\frac {{\left (x + \frac {c}{d}\right )} e^{\left (b d^{2} x^{2} \log \left (F\right ) + 2 \, b c d x \log \left (F\right ) + b c^{2} \log \left (F\right ) + a \log \left (F\right )\right )}}{2 \, b \log \left (F\right )} + \frac {\sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} d {\left (x + \frac {c}{d}\right )}\right )}{4 \, \sqrt {-b \log \left (F\right )} b d \log \left (F\right )} \]
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Time = 0.33 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.69 \[ \int F^{a+b (c+d x)^2} (c+d x)^2 \, dx=\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,x}{2\,b\,\ln \left (F\right )}-\frac {F^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \left (F\right )\,d^2+b\,c\,\ln \left (F\right )\,d}{\sqrt {b\,d^2\,\ln \left (F\right )}}\right )}{4\,b\,\ln \left (F\right )\,\sqrt {b\,d^2\,\ln \left (F\right )}}+\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,c}{2\,b\,d\,\ln \left (F\right )} \]
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