Optimal. Leaf size=78 \[ \frac{e^{c+d x^2} \text{Erfi}(a+b x)}{2 d}-\frac{b e^{\frac{a^2 d}{b^2+d}+c} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{2 d \sqrt{b^2+d}} \]
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Rubi [A] time = 0.0532274, antiderivative size = 78, 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 = {6384, 2234, 2204} \[ \frac{e^{c+d x^2} \text{Erfi}(a+b x)}{2 d}-\frac{b e^{\frac{a^2 d}{b^2+d}+c} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{2 d \sqrt{b^2+d}} \]
Antiderivative was successfully verified.
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Rule 6384
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int e^{c+d x^2} x \text{erfi}(a+b x) \, dx &=\frac{e^{c+d x^2} \text{erfi}(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{erfi}(a+b x)}{2 d}-\frac{\left (b e^{c+\frac{a^2 d}{b^2+d}}\right ) \int e^{\frac{\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{d \sqrt{\pi }}\\ &=\frac{e^{c+d x^2} \text{erfi}(a+b x)}{2 d}-\frac{b e^{c+\frac{a^2 d}{b^2+d}} \text{erfi}\left (\frac{a b+\left (b^2+d\right ) x}{\sqrt{b^2+d}}\right )}{2 d \sqrt{b^2+d}}\\ \end{align*}
Mathematica [A] time = 0.0707114, size = 73, normalized size = 0.94 \[ \frac{e^c \left (e^{d x^2} \text{Erfi}(a+b x)-\frac{b e^{\frac{a^2 d}{b^2+d}} \text{Erfi}\left (\frac{a b+x \left (b^2+d\right )}{\sqrt{b^2+d}}\right )}{\sqrt{b^2+d}}\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.261, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}x{\it erfi} \left ( bx+a \right ) \, 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 x \operatorname{erfi}\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 = 2.67521, size = 221, normalized size = 2.83 \begin{align*} \frac{\sqrt{-b^{2} - d} b \operatorname{erf}\left (\frac{{\left (a b +{\left (b^{2} + d\right )} x\right )} \sqrt{-b^{2} - d}}{b^{2} + d}\right ) e^{\left (\frac{b^{2} c +{\left (a^{2} + c\right )} d}{b^{2} + d}\right )} +{\left (b^{2} + d\right )} \operatorname{erfi}\left (b x + a\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{erfi}\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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