Optimal. Leaf size=57 \[ \frac {\text {erf}(b x) e^{c+d x^2}}{2 d}-\frac {b e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{2 d \sqrt {b^2-d}} \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6382, 2205} \[ \frac {\text {Erf}(b x) e^{c+d x^2}}{2 d}-\frac {b e^c \text {Erf}\left (x \sqrt {b^2-d}\right )}{2 d \sqrt {b^2-d}} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 6382
Rubi steps
\begin {align*} \int e^{c+d x^2} x \text {erf}(b x) \, dx &=\frac {e^{c+d x^2} \text {erf}(b x)}{2 d}-\frac {b \int e^{c-\left (b^2-d\right ) x^2} \, dx}{d \sqrt {\pi }}\\ &=\frac {e^{c+d x^2} \text {erf}(b x)}{2 d}-\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{2 \sqrt {b^2-d} d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 51, normalized size = 0.89 \[ \frac {e^c \left (e^{d x^2} \text {erf}(b x)-\frac {b \text {erfi}\left (x \sqrt {d-b^2}\right )}{\sqrt {d-b^2}}\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 62, normalized size = 1.09 \[ -\frac {\sqrt {b^{2} - d} b \operatorname {erf}\left (\sqrt {b^{2} - d} x\right ) e^{c} - {\left (b^{2} - d\right )} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{2 \, {\left (b^{2} d - d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 48, normalized size = 0.84 \[ \frac {b \operatorname {erf}\left (-\sqrt {b^{2} - d} x\right ) e^{c}}{2 \, \sqrt {b^{2} - d} d} + \frac {\operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 67, normalized size = 1.18 \[ \frac {\frac {\erf \left (b x \right ) b \,{\mathrm e}^{\frac {b^{2} d \,x^{2}+c \,b^{2}}{b^{2}}}}{2 d}-\frac {b \,{\mathrm e}^{c} \erf \left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{2 d \sqrt {1-\frac {d}{b^{2}}}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 47, normalized size = 0.82 \[ -\frac {b \operatorname {erf}\left (\sqrt {b^{2} - d} x\right ) e^{c}}{2 \, \sqrt {b^{2} - d} d} + \frac {\operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 47, normalized size = 0.82 \[ \frac {{\mathrm {e}}^{d\,x^2}\,{\mathrm {e}}^c\,\mathrm {erf}\left (b\,x\right )}{2\,d}-\frac {b\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {b^2-d}\right )}{2\,d\,\sqrt {b^2-d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int x e^{d x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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