Optimal. Leaf size=142 \[ \frac{b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{2 d^2 \sqrt{b^2+d}}+\frac{b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{4 d \left (b^2+d\right )^{3/2}}-\frac{b x e^{x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )}-\frac{\text{Erfi}(b x) e^{c+d x^2}}{2 d^2}+\frac{x^2 \text{Erfi}(b x) e^{c+d x^2}}{2 d} \]
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Rubi [A] time = 0.151724, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {6387, 6384, 2204, 2212} \[ \frac{b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{2 d^2 \sqrt{b^2+d}}+\frac{b e^c \text{Erfi}\left (x \sqrt{b^2+d}\right )}{4 d \left (b^2+d\right )^{3/2}}-\frac{b x e^{x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )}-\frac{\text{Erfi}(b x) e^{c+d x^2}}{2 d^2}+\frac{x^2 \text{Erfi}(b x) e^{c+d x^2}}{2 d} \]
Antiderivative was successfully verified.
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Rule 6387
Rule 6384
Rule 2204
Rule 2212
Rubi steps
\begin{align*} \int e^{c+d x^2} x^3 \text{erfi}(b x) \, dx &=\frac{e^{c+d x^2} x^2 \text{erfi}(b x)}{2 d}-\frac{\int e^{c+d x^2} x \text{erfi}(b x) \, dx}{d}-\frac{b \int e^{c+\left (b^2+d\right ) x^2} x^2 \, dx}{d \sqrt{\pi }}\\ &=-\frac{b e^{c+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{e^{c+d x^2} \text{erfi}(b x)}{2 d^2}+\frac{e^{c+d x^2} x^2 \text{erfi}(b x)}{2 d}+\frac{b \int e^{c+\left (b^2+d\right ) x^2} \, dx}{d^2 \sqrt{\pi }}+\frac{b \int e^{c+\left (b^2+d\right ) x^2} \, dx}{2 d \left (b^2+d\right ) \sqrt{\pi }}\\ &=-\frac{b e^{c+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{e^{c+d x^2} \text{erfi}(b x)}{2 d^2}+\frac{e^{c+d x^2} x^2 \text{erfi}(b x)}{2 d}+\frac{b e^c \text{erfi}\left (\sqrt{b^2+d} x\right )}{4 d \left (b^2+d\right )^{3/2}}+\frac{b e^c \text{erfi}\left (\sqrt{b^2+d} x\right )}{2 d^2 \sqrt{b^2+d}}\\ \end{align*}
Mathematica [A] time = 0.173665, size = 91, normalized size = 0.64 \[ \frac{e^c \left (\frac{\left (2 b^3+3 b d\right ) \text{Erfi}\left (x \sqrt{b^2+d}\right )}{\left (b^2+d\right )^{3/2}}-\frac{2 b d x e^{x^2 \left (b^2+d\right )}}{\sqrt{\pi } \left (b^2+d\right )}+2 e^{d x^2} \left (d x^2-1\right ) \text{Erfi}(b x)\right )}{4 d^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.24, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}{x}^{3}{\it erfi} \left ( bx \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^{3} \operatorname{erfi}\left (b x\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 = 2.72853, size = 332, normalized size = 2.34 \begin{align*} -\frac{\pi{\left (2 \, b^{3} + 3 \, b d\right )} \sqrt{-b^{2} - d} \operatorname{erf}\left (\sqrt{-b^{2} - d} x\right ) e^{c} + 2 \, \sqrt{\pi }{\left (b^{3} d + b d^{2}\right )} x e^{\left (b^{2} x^{2} + d x^{2} + c\right )} - 2 \,{\left (\pi{\left (b^{4} d + 2 \, b^{2} d^{2} + d^{3}\right )} x^{2} - \pi{\left (b^{4} + 2 \, b^{2} d + d^{2}\right )}\right )} \operatorname{erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{4 \, \pi{\left (b^{4} d^{2} + 2 \, b^{2} d^{3} + d^{4}\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^{3} \operatorname{erfi}\left (b x\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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