Optimal. Leaf size=37 \[ \frac {e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^c x}{\sqrt {\pi } b} \]
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Rubi [A] time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6382, 8} \[ \frac {e^{b^2 x^2+c} \text {Erf}(b x)}{2 b^2}-\frac {e^c x}{\sqrt {\pi } b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 6382
Rubi steps
\begin {align*} \int e^{c+b^2 x^2} x \text {erf}(b x) \, dx &=\frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 b^2}-\frac {\int e^c \, dx}{b \sqrt {\pi }}\\ &=-\frac {e^c x}{b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 34, normalized size = 0.92 \[ \frac {e^c \left (e^{b^2 x^2} \text {erf}(b x)-\frac {2 b x}{\sqrt {\pi }}\right )}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 35, normalized size = 0.95 \[ -\frac {2 \, \sqrt {\pi } b x e^{c} - \pi \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, \pi b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 31, normalized size = 0.84 \[ -\frac {x e^{c}}{\sqrt {\pi } b} + \frac {\operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 51, normalized size = 1.38 \[ \frac {-2 \,{\mathrm e}^{b^{2} x^{2}+c} {\mathrm e}^{-b^{2} x^{2}} x b +{\mathrm e}^{b^{2} x^{2}+c} \erf \left (b x \right ) \sqrt {\pi }}{2 \sqrt {\pi }\, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 34, normalized size = 0.92 \[ -\frac {2 \, b x e^{c} - \sqrt {\pi } \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{2 \, \sqrt {\pi } b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 31, normalized size = 0.84 \[ \frac {{\mathrm {e}}^{b^2\,x^2}\,{\mathrm {e}}^c\,\mathrm {erf}\left (b\,x\right )}{2\,b^2}-\frac {x\,{\mathrm {e}}^c}{b\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.84, size = 34, normalized size = 0.92 \[ \begin {cases} - \frac {x e^{c}}{\sqrt {\pi } b} + \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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