Optimal. Leaf size=71 \[ \frac{2 b^3 e^c x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},b^2 x^2\right )}{\sqrt{\pi }}-\frac{e^{b^2 x^2+c} \text{Erf}(b x)}{2 x^2}-\frac{b e^c}{\sqrt{\pi } x} \]
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Rubi [A] time = 0.0891348, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {6391, 6388, 12, 30} \[ \frac{2 b^3 e^c x \, _2F_2\left (\frac{1}{2},1;\frac{3}{2},\frac{3}{2};b^2 x^2\right )}{\sqrt{\pi }}-\frac{e^{b^2 x^2+c} \text{Erf}(b x)}{2 x^2}-\frac{b e^c}{\sqrt{\pi } x} \]
Antiderivative was successfully verified.
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Rule 6391
Rule 6388
Rule 12
Rule 30
Rubi steps
\begin{align*} \int \frac{e^{c+b^2 x^2} \text{erf}(b x)}{x^3} \, dx &=-\frac{e^{c+b^2 x^2} \text{erf}(b x)}{2 x^2}+b^2 \int \frac{e^{c+b^2 x^2} \text{erf}(b x)}{x} \, dx+\frac{b \int \frac{e^c}{x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{e^{c+b^2 x^2} \text{erf}(b x)}{2 x^2}+\frac{2 b^3 e^c x \, _2F_2\left (\frac{1}{2},1;\frac{3}{2},\frac{3}{2};b^2 x^2\right )}{\sqrt{\pi }}+\frac{\left (b e^c\right ) \int \frac{1}{x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{b e^c}{\sqrt{\pi } x}-\frac{e^{c+b^2 x^2} \text{erf}(b x)}{2 x^2}+\frac{2 b^3 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 = 0.127304, size = 34, normalized size = 0.48 \[ -\frac{2 b e^c \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},1\right \},\left \{\frac{1}{2},\frac{3}{2}\right \},b^2 x^2\right )}{\sqrt{\pi } x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.267, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it Erf} \left ( bx \right ) }{{x}^{3}}}\, 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{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}}\,{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{\operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}}, 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{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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