Optimal. Leaf size=134 \[ -\frac{b^5 e^c x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},b^2 x^2\right )}{\sqrt{\pi }}-\frac{b^2 e^{b^2 x^2+c} \text{Erfc}(b x)}{4 x^2}-\frac{e^{b^2 x^2+c} \text{Erfc}(b x)}{4 x^4}+\frac{1}{4} b^4 e^c \text{ExpIntegralEi}\left (b^2 x^2\right )+\frac{b^3 e^c}{2 \sqrt{\pi } x}+\frac{b e^c}{6 \sqrt{\pi } x^3} \]
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Rubi [A] time = 0.201698, antiderivative size = 134, normalized size of antiderivative = 1., 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 6392
Rule 6389
Rule 2210
Rule 6388
Rule 12
Rule 30
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*}
Mathematica [A] time = 0.223616, size = 83, normalized size = 0.62 \[ -\frac{e^c \left (3 \sqrt{\pi } \left (e^{b^2 x^2} \left (b^2 x^2+1\right )-b^4 x^4 \text{ExpIntegralEi}\left (b^2 x^2\right )\right )-8 b x \text{HypergeometricPFQ}\left (\left \{-\frac{3}{2},1\right \},\left \{-\frac{1}{2},\frac{3}{2}\right \},b^2 x^2\right )\right )}{12 \sqrt{\pi } x^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.311, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it erfc} \left ( bx \right ) }{{x}^{5}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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