Integrand size = 17, antiderivative size = 155 \[ \int e^{c+d x^2} x^3 \text {erf}(b x) \, dx=\frac {b e^{c-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erf}(b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erf}(b x)}{2 d}+\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{2 \sqrt {b^2-d} d^2}-\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{4 \left (b^2-d\right )^{3/2} d} \]
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Time = 0.12 (sec) , antiderivative size = 155, 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 = {6520, 6517, 2236, 2243} \[ \int e^{c+d x^2} x^3 \text {erf}(b x) \, dx=\frac {b e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{2 d^2 \sqrt {b^2-d}}-\frac {b e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 d \left (b^2-d\right )^{3/2}}+\frac {b x e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt {\pi } d \left (b^2-d\right )}-\frac {\text {erf}(b x) e^{c+d x^2}}{2 d^2}+\frac {x^2 \text {erf}(b x) e^{c+d x^2}}{2 d} \]
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Rule 2236
Rule 2243
Rule 6517
Rule 6520
Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+d x^2} x^2 \text {erf}(b x)}{2 d}-\frac {\int e^{c+d x^2} x \text {erf}(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 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erf}(b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erf}(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 \left (b^2-d\right ) d \sqrt {\pi }} \\ & = \frac {b e^{c-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erf}(b x)}{2 d^2}+\frac {e^{c+d x^2} 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^2}-\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{4 \left (b^2-d\right )^{3/2} d} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.64 \[ \int e^{c+d x^2} x^3 \text {erf}(b x) \, dx=\frac {e^c \left (\frac {2 b d e^{\left (-b^2+d\right ) x^2} x}{\left (b^2-d\right ) \sqrt {\pi }}+2 e^{d x^2} \left (-1+d x^2\right ) \text {erf}(b x)+\frac {b \left (-2 b^2+3 d\right ) \text {erfi}\left (\sqrt {-b^2+d} x\right )}{\left (-b^2+d\right )^{3/2}}\right )}{4 d^2} \]
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Time = 1.44 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{c} \left (\frac {b^{4} x^{2} {\mathrm e}^{d \,x^{2}}}{2 d}-\frac {b^{4} {\mathrm e}^{d \,x^{2}}}{2 d^{2}}\right )}{b^{3}}-\frac {{\mathrm e}^{c} \left (\frac {b^{2} \left (\frac {b x \,{\mathrm e}^{\left (-1+\frac {d}{b^{2}}\right ) b^{2} x^{2}}}{-2+\frac {2 d}{b^{2}}}-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{4 \left (-1+\frac {d}{b^{2}}\right ) \sqrt {1-\frac {d}{b^{2}}}}\right )}{d}-\frac {b^{4} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{2 d^{2} \sqrt {1-\frac {d}{b^{2}}}}\right )}{\sqrt {\pi }\, b^{3}}}{b}\) | \(168\) |
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Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.96 \[ \int e^{c+d x^2} x^3 \text {erf}(b x) \, dx=\frac {\pi {\left (2 \, b^{3} - 3 \, b d\right )} \sqrt {b^{2} - d} \operatorname {erf}\left (\sqrt {b^{2} - d} x\right ) e^{c} + 2 \, \sqrt {\pi } {\left (b^{3} d - b d^{2}\right )} x e^{\left (-b^{2} x^{2} + d x^{2} + c\right )} + 2 \, {\left (\pi {\left (b^{4} d - 2 \, b^{2} d^{2} + d^{3}\right )} x^{2} - \pi {\left (b^{4} - 2 \, b^{2} d + d^{2}\right )}\right )} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{4 \, \pi {\left (b^{4} d^{2} - 2 \, b^{2} d^{3} + d^{4}\right )}} \]
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\[ \int e^{c+d x^2} x^3 \text {erf}(b x) \, dx=e^{c} \int x^{3} e^{d x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]
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\[ \int e^{c+d x^2} x^3 \text {erf}(b x) \, dx=\int { x^{3} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.91 \[ \int e^{c+d x^2} x^3 \text {erf}(b x) \, dx=\frac {1}{2} \, {\left (\frac {{\left (d x^{2} + c - 1\right )} e^{\left (d x^{2} + c\right )}}{d^{2}} - \frac {c e^{\left (d x^{2} + c\right )}}{d^{2}}\right )} \operatorname {erf}\left (b x\right ) + \frac {b d {\left (\frac {2 \, x e^{\left (-b^{2} x^{2} + d x^{2} + c\right )}}{b^{2} - d} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {b^{2} - d} x\right ) e^{c}}{{\left (b^{2} - d\right )}^{\frac {3}{2}}}\right )} - \frac {2 \, \sqrt {\pi } b \operatorname {erf}\left (-\sqrt {b^{2} - d} x\right ) e^{c}}{\sqrt {b^{2} - d}}}{4 \, \sqrt {\pi } d^{2}} \]
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Time = 0.91 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.85 \[ \int e^{c+d x^2} x^3 \text {erf}(b x) \, dx=\frac {b\,x\,{\mathrm {e}}^{-b^2\,x^2+d\,x^2+c}}{2\,\sqrt {\pi }\,\left (b^2\,d-d^2\right )}-\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{d\,x^2+c}}{2\,d^2}-\frac {x^2\,{\mathrm {e}}^{d\,x^2+c}}{2\,d}\right )+\frac {b\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {b^2-d}\right )}{2\,d^2\,\sqrt {b^2-d}}+\frac {b\,{\mathrm {e}}^c\,\mathrm {erfi}\left (x\,\sqrt {d-b^2}\right )}{4\,d\,{\left (d-b^2\right )}^{3/2}} \]
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