Integrand size = 17, antiderivative size = 17 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^5} \, dx=\frac {b e^{c-\left (b^2-d\right ) x^2}}{6 \sqrt {\pi } x^3}-\frac {b \left (b^2-d\right ) e^{c-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}+\frac {b d e^{c-\left (b^2-d\right ) x^2}}{2 \sqrt {\pi } x}-\frac {1}{3} b \left (b^2-d\right )^{3/2} e^c \text {erf}\left (\sqrt {b^2-d} x\right )+\frac {1}{2} b \sqrt {b^2-d} d e^c \text {erf}\left (\sqrt {b^2-d} x\right )-\frac {e^{c+d x^2} \text {erfc}(b x)}{4 x^4}-\frac {d e^{c+d x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{2} d^2 \text {Int}\left (\frac {e^{c+d x^2} \text {erfc}(b x)}{x},x\right ) \]
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Not integrable
Time = 0.22 (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 = {} \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^5} \, dx=\int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^5} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {e^{c+d x^2} \text {erfc}(b x)}{4 x^4}+\frac {1}{2} d \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^3} \, dx-\frac {b \int \frac {e^{c-\left (b^2-d\right ) x^2}}{x^4} \, dx}{2 \sqrt {\pi }} \\ & = \frac {b e^{c-\left (b^2-d\right ) x^2}}{6 \sqrt {\pi } x^3}-\frac {e^{c+d x^2} \text {erfc}(b x)}{4 x^4}-\frac {d e^{c+d x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{2} d^2 \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x} \, dx+\frac {\left (b \left (b^2-d\right )\right ) \int \frac {e^{c+\left (-b^2+d\right ) x^2}}{x^2} \, dx}{3 \sqrt {\pi }}-\frac {(b d) \int \frac {e^{c-\left (b^2-d\right ) x^2}}{x^2} \, dx}{2 \sqrt {\pi }} \\ & = \frac {b e^{c-\left (b^2-d\right ) x^2}}{6 \sqrt {\pi } x^3}-\frac {b \left (b^2-d\right ) e^{c-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}+\frac {b d e^{c-\left (b^2-d\right ) x^2}}{2 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfc}(b x)}{4 x^4}-\frac {d e^{c+d x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{2} d^2 \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x} \, dx-\frac {\left (2 b \left (b^2-d\right )^2\right ) \int e^{c+\left (-b^2+d\right ) x^2} \, dx}{3 \sqrt {\pi }}+\frac {\left (b \left (b^2-d\right ) d\right ) \int e^{c+\left (-b^2+d\right ) x^2} \, dx}{\sqrt {\pi }} \\ & = \frac {b e^{c-\left (b^2-d\right ) x^2}}{6 \sqrt {\pi } x^3}-\frac {b \left (b^2-d\right ) e^{c-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}+\frac {b d e^{c-\left (b^2-d\right ) x^2}}{2 \sqrt {\pi } x}-\frac {1}{3} b \left (b^2-d\right )^{3/2} e^c \text {erf}\left (\sqrt {b^2-d} x\right )+\frac {1}{2} b \sqrt {b^2-d} d e^c \text {erf}\left (\sqrt {b^2-d} x\right )-\frac {e^{c+d x^2} \text {erfc}(b x)}{4 x^4}-\frac {d e^{c+d x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{2} d^2 \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x} \, dx \\ \end{align*}
Not integrable
Time = 0.55 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^5} \, dx=\int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^5} \, dx \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
\[\int \frac {{\mathrm e}^{d \,x^{2}+c} \operatorname {erfc}\left (b x \right )}{x^{5}}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{5}} \,d x } \]
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Not integrable
Time = 32.35 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^5} \, dx=e^{c} \int \frac {e^{d x^{2}} \operatorname {erfc}{\left (b x \right )}}{x^{5}}\, dx \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{5}} \,d x } \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{5}} \,d x } \]
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Not integrable
Time = 4.83 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfc}(b x)}{x^5} \, dx=\int \frac {{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x^5} \,d x \]
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