Optimal. Leaf size=86 \[ \frac{b e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{2 d \sqrt{b^2-d}}+\frac{e^{c+d x^2} \text{Erfc}(a+b x)}{2 d} \]
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Rubi [A] time = 0.0535091, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {6383, 2234, 2205} \[ \frac{b e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{2 d \sqrt{b^2-d}}+\frac{e^{c+d x^2} \text{Erfc}(a+b x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 6383
Rule 2234
Rule 2205
Rubi steps
\begin{align*} \int e^{c+d x^2} x \text{erfc}(a+b x) \, dx &=\frac{e^{c+d x^2} \text{erfc}(a+b x)}{2 d}+\frac{b \int e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} \, dx}{d \sqrt{\pi }}\\ &=\frac{e^{c+d x^2} \text{erfc}(a+b x)}{2 d}+\frac{\left (b e^{\frac{b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac{\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{d \sqrt{\pi }}\\ &=\frac{b e^{\frac{b^2 c+a^2 d-c d}{b^2-d}} \text{erf}\left (\frac{a b+\left (b^2-d\right ) x}{\sqrt{b^2-d}}\right )}{2 \sqrt{b^2-d} d}+\frac{e^{c+d x^2} \text{erfc}(a+b x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.105111, size = 81, normalized size = 0.94 \[ \frac{e^c \left (\frac{b e^{\frac{a^2 d}{b^2-d}} \text{Erfi}\left (\frac{x \left (d-b^2\right )-a b}{\sqrt{d-b^2}}\right )}{\sqrt{d-b^2}}+e^{d x^2} \text{Erfc}(a+b x)\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.361, size = 175, normalized size = 2. \begin{align*}{\frac{1}{b} \left ({\frac{b}{2\,d}{{\rm e}^{{\frac{d \left ( bx+a \right ) ^{2}}{{b}^{2}}}-2\,{\frac{ad \left ( bx+a \right ) }{{b}^{2}}}+{\frac{{a}^{2}d}{{b}^{2}}}+c}}}-{\frac{{\it Erf} \left ( bx+a \right ) b}{2\,d}{{\rm e}^{{\frac{d \left ( bx+a \right ) ^{2}}{{b}^{2}}}-2\,{\frac{ad \left ( bx+a \right ) }{{b}^{2}}}+{\frac{{a}^{2}d}{{b}^{2}}}+c}}}+{\frac{b}{2\,d}{{\rm e}^{{\frac{{a}^{2}d}{{b}^{2}}}+c-{\frac{{a}^{2}{d}^{2}}{{b}^{4}} \left ( -1+{\frac{d}{{b}^{2}}} \right ) ^{-1}}}}{\it Erf} \left ( \sqrt{1-{\frac{d}{{b}^{2}}}} \left ( bx+a \right ) +{\frac{ad}{{b}^{2}}{\frac{1}{\sqrt{1-{\frac{d}{{b}^{2}}}}}}} \right ){\frac{1}{\sqrt{1-{\frac{d}{{b}^{2}}}}}}} \right ) } \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 x \operatorname{erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.45632, size = 220, normalized size = 2.56 \begin{align*} \frac{\sqrt{b^{2} - d} b \operatorname{erf}\left (\frac{a b +{\left (b^{2} - d\right )} x}{\sqrt{b^{2} - d}}\right ) e^{\left (\frac{b^{2} c +{\left (a^{2} - c\right )} d}{b^{2} - d}\right )} +{\left (b^{2} -{\left (b^{2} - d\right )} \operatorname{erf}\left (b x + a\right ) - d\right )} e^{\left (d x^{2} + c\right )}}{2 \,{\left (b^{2} d - d^{2}\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 x \operatorname{erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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