Integrand size = 19, antiderivative size = 342 \[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {b e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \sqrt {b^2-d} d^2}+\frac {a^2 b^3 e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{5/2} d}+\frac {b e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2} d}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfc}(a+b x)}{2 d} \]
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Time = 0.32 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6521, 6518, 2266, 2236, 2273, 2272} \[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=-\frac {b e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d^2 \sqrt {b^2-d}}+\frac {b e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 d \left (b^2-d\right )^{3/2}}+\frac {a b^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt {\pi } d \left (b^2-d\right )^2}-\frac {b x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt {\pi } d \left (b^2-d\right )}+\frac {a^2 b^3 e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d \left (b^2-d\right )^{5/2}}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d^2}+\frac {x^2 e^{c+d x^2} \text {erfc}(a+b x)}{2 d} \]
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Rule 2236
Rule 2266
Rule 2272
Rule 2273
Rule 6518
Rule 6521
Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+d x^2} x^2 \text {erfc}(a+b x)}{2 d}-\frac {\int e^{c+d x^2} x \text {erfc}(a+b x) \, dx}{d}+\frac {b \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} x^2 \, dx}{d \sqrt {\pi }} \\ & = -\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfc}(a+b x)}{2 d}-\frac {b \int e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} \, dx}{d^2 \sqrt {\pi }}+\frac {b \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {\left (a b^2\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} x \, dx}{\left (b^2-d\right ) d \sqrt {\pi }} \\ & = \frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfc}(a+b x)}{2 d}+\frac {\left (a^2 b^3\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{\left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {\left (b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{d^2 \sqrt {\pi }}+\frac {\left (b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{2 \left (b^2-d\right ) d \sqrt {\pi }} \\ & = \frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \sqrt {b^2-d} d^2}+\frac {b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2} d}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfc}(a+b x)}{2 d}+\frac {\left (a^2 b^3 e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{\left (b^2-d\right )^2 d \sqrt {\pi }} \\ & = \frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \sqrt {b^2-d} d^2}+\frac {a^2 b^3 e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{5/2} d}+\frac {b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2} d}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfc}(a+b x)}{2 d} \\ \end{align*}
Time = 3.22 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.75 \[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=-\frac {e^c \left (-2 e^{d x^2} \left (-1+d x^2\right )+2 e^{d x^2} \left (-1+d x^2\right ) \text {erf}(a+b x)-\frac {b d e^{-a^2-2 a b x+\left (-b^2+d\right ) x^2} \left (2 \left (b^2-d\right ) \left (a b+\left (-b^2+d\right ) x\right )+\sqrt {b^2-d} \left (\left (1+2 a^2\right ) b^2-d\right ) e^{\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}} \sqrt {\pi } \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )\right )}{\left (b^2-d\right )^3 \sqrt {\pi }}+\frac {2 b e^{\frac {a^2 d}{b^2-d}} \text {erfi}\left (\frac {-a b+\left (-b^2+d\right ) x}{\sqrt {-b^2+d}}\right )}{\sqrt {-b^2+d}}\right )}{4 d^2} \]
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\[\int {\mathrm e}^{d \,x^{2}+c} x^{3} \operatorname {erfc}\left (b x +a \right )d x\]
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Time = 0.27 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.96 \[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=-\frac {\pi {\left (2 \, b^{5} - {\left (2 \, a^{2} + 5\right )} b^{3} d + 3 \, b d^{2}\right )} \sqrt {b^{2} - d} \operatorname {erf}\left (\frac {a b + {\left (b^{2} - d\right )} x}{\sqrt {b^{2} - d}}\right ) e^{\left (\frac {b^{2} c + {\left (a^{2} - c\right )} d}{b^{2} - d}\right )} - 2 \, \sqrt {\pi } {\left (a b^{4} d - a b^{2} d^{2} - {\left (b^{5} d - 2 \, b^{3} d^{2} + b d^{3}\right )} x\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x + d x^{2} - a^{2} + c\right )} - 2 \, {\left (\pi {\left (b^{6} d - 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} - d^{4}\right )} x^{2} - \pi {\left (b^{6} - 3 \, b^{4} d + 3 \, b^{2} d^{2} - d^{3}\right )} - {\left (\pi {\left (b^{6} d - 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} - d^{4}\right )} x^{2} - \pi {\left (b^{6} - 3 \, b^{4} d + 3 \, b^{2} d^{2} - d^{3}\right )}\right )} \operatorname {erf}\left (b x + a\right )\right )} e^{\left (d x^{2} + c\right )}}{4 \, \pi {\left (b^{6} d^{2} - 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} - d^{5}\right )}} \]
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\[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=e^{c} \int x^{3} e^{d x^{2}} \operatorname {erfc}{\left (a + b x \right )}\, dx \]
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\[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
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\[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=\int { x^{3} \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
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Timed out. \[ \int e^{c+d x^2} x^3 \text {erfc}(a+b x) \, dx=\int x^3\,\mathrm {erfc}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c} \,d x \]
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