Integrand size = 17, antiderivative size = 142 \[ \int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx=-\frac {b e^{c+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erfi}(b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfi}(b x)}{2 d}+\frac {b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{4 d \left (b^2+d\right )^{3/2}}+\frac {b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{2 d^2 \sqrt {b^2+d}} \]
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Time = 0.11 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6522, 6519, 2235, 2243} \[ \int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx=\frac {b e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{2 d^2 \sqrt {b^2+d}}+\frac {b e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{4 d \left (b^2+d\right )^{3/2}}-\frac {b x e^{x^2 \left (b^2+d\right )+c}}{2 \sqrt {\pi } d \left (b^2+d\right )}-\frac {\text {erfi}(b x) e^{c+d x^2}}{2 d^2}+\frac {x^2 \text {erfi}(b x) e^{c+d x^2}}{2 d} \]
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Rule 2235
Rule 2243
Rule 6519
Rule 6522
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.64 \[ \int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx=\frac {e^c \left (-\frac {2 b d e^{\left (b^2+d\right ) x^2} x}{\left (b^2+d\right ) \sqrt {\pi }}+2 e^{d x^2} \left (-1+d x^2\right ) \text {erfi}(b x)+\frac {\left (2 b^3+3 b d\right ) \text {erfi}\left (\sqrt {b^2+d} x\right )}{\left (b^2+d\right )^{3/2}}\right )}{4 d^2} \]
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\[\int {\mathrm e}^{d \,x^{2}+c} x^{3} \operatorname {erfi}\left (b x \right )d x\]
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Time = 0.27 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.06 \[ \int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx=-\frac {\pi {\left (2 \, b^{3} + 3 \, b d\right )} \sqrt {-b^{2} - d} \operatorname {erf}\left (\sqrt {-b^{2} - d} x\right ) e^{c} + 2 \, \sqrt {\pi } {\left (b^{3} d + b d^{2}\right )} x e^{\left (b^{2} x^{2} + d x^{2} + c\right )} - 2 \, {\left (\pi {\left (b^{4} d + 2 \, b^{2} d^{2} + d^{3}\right )} x^{2} - \pi {\left (b^{4} + 2 \, b^{2} d + d^{2}\right )}\right )} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{4 \, \pi {\left (b^{4} d^{2} + 2 \, b^{2} d^{3} + d^{4}\right )}} \]
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\[ \int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx=e^{c} \int x^{3} e^{d x^{2}} \operatorname {erfi}{\left (b x \right )}\, dx \]
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\[ \int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx=\int { x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
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\[ \int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx=\int { x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
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Time = 5.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.90 \[ \int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx=\frac {b\,\mathrm {erfi}\left (x\,\sqrt {b^2+d}\right )\,{\mathrm {e}}^c}{4\,d\,{\left (b^2+d\right )}^{3/2}}-\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{d\,x^2+c}}{2\,d^2}-\frac {x^2\,{\mathrm {e}}^{d\,x^2+c}}{2\,d}\right )-\frac {b\,x\,{\mathrm {e}}^{b^2\,x^2+d\,x^2+c}}{2\,\sqrt {\pi }\,\left (b^2\,d+d^2\right )}+\frac {b\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {-b^2-d}\right )}{2\,d^2\,\sqrt {-b^2-d}} \]
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