Optimal. Leaf size=71 \[ \frac {2 b^3 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{2 x^2}-\frac {b e^c}{\sqrt {\pi } x} \]
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Rubi [A] time = 0.09, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, 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 12
Rule 30
Rule 6388
Rule 6391
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*}
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Mathematica [A] time = 0.13, size = 34, normalized size = 0.48 \[ -\frac {2 b e^c \, _2F_2\left (-\frac {1}{2},1;\frac {1}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi } x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}}, x\right ) \]
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 \[ \int \frac {\operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{b^{2} x^{2}+c} \erf \left (b x \right )}{x^{3}}\, dx \]
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 \[ \int \frac {\operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erf}\left (b\,x\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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