Integrand size = 17, antiderivative size = 17 \[ \int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx=\frac {3 b e^{c-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {b e^{c-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{c-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erfc}(b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(b x)}{2 d}+\frac {3 \text {Int}\left (e^{c+d x^2} \text {erfc}(b x),x\right )}{4 d^2} \]
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Not integrable
Time = 0.17 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx=\int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+d x^2} x^3 \text {erfc}(b x)}{2 d}-\frac {3 \int e^{c+d x^2} x^2 \text {erfc}(b x) \, dx}{2 d}+\frac {b \int e^{c-\left (b^2-d\right ) x^2} x^3 \, dx}{d \sqrt {\pi }} \\ & = -\frac {b e^{c-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erfc}(b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfc}(b x) \, dx}{4 d^2}-\frac {(3 b) \int e^{c-\left (b^2-d\right ) x^2} x \, dx}{2 d^2 \sqrt {\pi }}+\frac {b \int e^{c+\left (-b^2+d\right ) x^2} x \, dx}{\left (b^2-d\right ) d \sqrt {\pi }} \\ & = \frac {3 b e^{c-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {b e^{c-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{c-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erfc}(b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfc}(b x) \, dx}{4 d^2} \\ \end{align*}
Not integrable
Time = 0.58 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx=\int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx \]
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Not integrable
Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
\[\int {\mathrm e}^{d \,x^{2}+c} x^{4} \operatorname {erfc}\left (b x \right )d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
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Not integrable
Time = 41.92 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx=e^{c} \int x^{4} e^{d x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
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Not integrable
Time = 4.89 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx=\int x^4\,{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]
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