Optimal. Leaf size=134 \[ -\frac {b^5 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}+\frac {b^3 e^c}{2 \sqrt {\pi } x}-\frac {b^2 e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^2}-\frac {e^{b^2 x^2+c} \text {erfc}(b x)}{4 x^4}+\frac {1}{4} b^4 e^c \text {Ei}\left (b^2 x^2\right )+\frac {b e^c}{6 \sqrt {\pi } x^3} \]
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Rubi [A] time = 0.20, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6392, 6389, 2210, 6388, 12, 30} \[ -\frac {b^5 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}-\frac {b^2 e^{b^2 x^2+c} \text {Erfc}(b x)}{4 x^2}-\frac {e^{b^2 x^2+c} \text {Erfc}(b x)}{4 x^4}+\frac {1}{4} b^4 e^c \text {Ei}\left (b^2 x^2\right )+\frac {b^3 e^c}{2 \sqrt {\pi } x}+\frac {b e^c}{6 \sqrt {\pi } x^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2210
Rule 6388
Rule 6389
Rule 6392
Rubi steps
\begin {align*} \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^5} \, dx &=-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^4}+\frac {1}{2} b^2 \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^3} \, dx-\frac {b \int \frac {e^c}{x^4} \, dx}{2 \sqrt {\pi }}\\ &=-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x} \, dx-\frac {b^3 \int \frac {e^c}{x^2} \, dx}{2 \sqrt {\pi }}-\frac {\left (b e^c\right ) \int \frac {1}{x^4} \, dx}{2 \sqrt {\pi }}\\ &=\frac {b e^c}{6 \sqrt {\pi } x^3}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{2} b^4 \int \frac {e^{c+b^2 x^2}}{x} \, dx-\frac {1}{2} b^4 \int \frac {e^{c+b^2 x^2} \text {erf}(b x)}{x} \, dx-\frac {\left (b^3 e^c\right ) \int \frac {1}{x^2} \, dx}{2 \sqrt {\pi }}\\ &=\frac {b e^c}{6 \sqrt {\pi } x^3}+\frac {b^3 e^c}{2 \sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^4}-\frac {b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{4 x^2}+\frac {1}{4} b^4 e^c \text {Ei}\left (b^2 x^2\right )-\frac {b^5 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 83, normalized size = 0.62 \[ -\frac {e^c \left (3 \sqrt {\pi } \left (e^{b^2 x^2} \left (b^2 x^2+1\right )-b^4 x^4 \text {Ei}\left (b^2 x^2\right )\right )-8 b x \, _2F_2\left (-\frac {3}{2},1;-\frac {1}{2},\frac {3}{2};b^2 x^2\right )\right )}{12 \sqrt {\pi } x^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\operatorname {erf}\left (b x\right ) - 1\right )} e^{\left (b^{2} x^{2} + c\right )}}{x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{b^{2} x^{2}+c} \mathrm {erfc}\left (b x \right )}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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