Integrand size = 17, antiderivative size = 257 \[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=\frac {3 b e^{c+\left (b^2+d\right ) x^2} x}{4 d \left (b^2+d\right )^2 \sqrt {\pi }}+\frac {b e^{c+\left (b^2+d\right ) x^2} x}{d^2 \left (b^2+d\right ) \sqrt {\pi }}-\frac {b e^{c+\left (b^2+d\right ) x^2} x^3}{2 d \left (b^2+d\right ) \sqrt {\pi }}+\frac {e^{c+d x^2} \text {erfi}(b x)}{d^3}-\frac {e^{c+d x^2} x^2 \text {erfi}(b x)}{d^2}+\frac {e^{c+d x^2} x^4 \text {erfi}(b x)}{2 d}-\frac {3 b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{8 d \left (b^2+d\right )^{5/2}}-\frac {b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{2 d^2 \left (b^2+d\right )^{3/2}}-\frac {b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{d^3 \sqrt {b^2+d}} \]
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Time = 0.32 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6522, 6519, 2235, 2243} \[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=-\frac {b e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{d^3 \sqrt {b^2+d}}-\frac {b e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{2 d^2 \left (b^2+d\right )^{3/2}}+\frac {b x e^{x^2 \left (b^2+d\right )+c}}{\sqrt {\pi } d^2 \left (b^2+d\right )}-\frac {3 b e^c \text {erfi}\left (x \sqrt {b^2+d}\right )}{8 d \left (b^2+d\right )^{5/2}}+\frac {3 b x e^{x^2 \left (b^2+d\right )+c}}{4 \sqrt {\pi } d \left (b^2+d\right )^2}-\frac {b x^3 e^{x^2 \left (b^2+d\right )+c}}{2 \sqrt {\pi } d \left (b^2+d\right )}+\frac {\text {erfi}(b x) e^{c+d x^2}}{d^3}-\frac {x^2 \text {erfi}(b x) e^{c+d x^2}}{d^2}+\frac {x^4 \text {erfi}(b x) e^{c+d x^2}}{2 d} \]
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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^4 \text {erfi}(b x)}{2 d}-\frac {2 \int e^{c+d x^2} x^3 \text {erfi}(b x) \, dx}{d}-\frac {b \int e^{c+\left (b^2+d\right ) x^2} x^4 \, dx}{d \sqrt {\pi }} \\ & = -\frac {b e^{c+\left (b^2+d\right ) x^2} x^3}{2 d \left (b^2+d\right ) \sqrt {\pi }}-\frac {e^{c+d x^2} x^2 \text {erfi}(b x)}{d^2}+\frac {e^{c+d x^2} x^4 \text {erfi}(b x)}{2 d}+\frac {2 \int e^{c+d x^2} x \text {erfi}(b x) \, dx}{d^2}+\frac {(2 b) \int e^{c+\left (b^2+d\right ) x^2} x^2 \, dx}{d^2 \sqrt {\pi }}+\frac {(3 b) \int e^{c+\left (b^2+d\right ) x^2} x^2 \, dx}{2 d \left (b^2+d\right ) \sqrt {\pi }} \\ & = \frac {3 b e^{c+\left (b^2+d\right ) x^2} x}{4 d \left (b^2+d\right )^2 \sqrt {\pi }}+\frac {b e^{c+\left (b^2+d\right ) x^2} x}{d^2 \left (b^2+d\right ) \sqrt {\pi }}-\frac {b e^{c+\left (b^2+d\right ) x^2} x^3}{2 d \left (b^2+d\right ) \sqrt {\pi }}+\frac {e^{c+d x^2} \text {erfi}(b x)}{d^3}-\frac {e^{c+d x^2} x^2 \text {erfi}(b x)}{d^2}+\frac {e^{c+d x^2} x^4 \text {erfi}(b x)}{2 d}-\frac {(2 b) \int e^{c+\left (b^2+d\right ) x^2} \, dx}{d^3 \sqrt {\pi }}-\frac {(3 b) \int e^{c+\left (b^2+d\right ) x^2} \, dx}{4 d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {b \int e^{c+\left (b^2+d\right ) x^2} \, dx}{d^2 \left (b^2+d\right ) \sqrt {\pi }} \\ & = \frac {3 b e^{c+\left (b^2+d\right ) x^2} x}{4 d \left (b^2+d\right )^2 \sqrt {\pi }}+\frac {b e^{c+\left (b^2+d\right ) x^2} x}{d^2 \left (b^2+d\right ) \sqrt {\pi }}-\frac {b e^{c+\left (b^2+d\right ) x^2} x^3}{2 d \left (b^2+d\right ) \sqrt {\pi }}+\frac {e^{c+d x^2} \text {erfi}(b x)}{d^3}-\frac {e^{c+d x^2} x^2 \text {erfi}(b x)}{d^2}+\frac {e^{c+d x^2} x^4 \text {erfi}(b x)}{2 d}-\frac {3 b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{8 d \left (b^2+d\right )^{5/2}}-\frac {b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{2 d^2 \left (b^2+d\right )^{3/2}}-\frac {b e^c \text {erfi}\left (\sqrt {b^2+d} x\right )}{d^3 \sqrt {b^2+d}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.51 \[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=\frac {e^c \left (-\frac {2 b d e^{\left (b^2+d\right ) x^2} x \left (2 b^2 \left (-2+d x^2\right )+d \left (-7+2 d x^2\right )\right )}{\left (b^2+d\right )^2 \sqrt {\pi }}+4 e^{d x^2} \left (2-2 d x^2+d^2 x^4\right ) \text {erfi}(b x)-\frac {b \left (8 b^4+20 b^2 d+15 d^2\right ) \text {erfi}\left (\sqrt {b^2+d} x\right )}{\left (b^2+d\right )^{5/2}}\right )}{8 d^3} \]
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\[\int {\mathrm e}^{d \,x^{2}+c} x^{5} \operatorname {erfi}\left (b x \right )d x\]
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Time = 0.27 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.99 \[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=\frac {\pi {\left (8 \, b^{5} + 20 \, b^{3} d + 15 \, b d^{2}\right )} \sqrt {-b^{2} - d} \operatorname {erf}\left (\sqrt {-b^{2} - d} x\right ) e^{c} + 4 \, {\left (\pi {\left (b^{6} d^{2} + 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} + d^{5}\right )} x^{4} - 2 \, \pi {\left (b^{6} d + 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} + d^{4}\right )} x^{2} + 2 \, \pi {\left (b^{6} + 3 \, b^{4} d + 3 \, b^{2} d^{2} + d^{3}\right )}\right )} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )} - 2 \, \sqrt {\pi } {\left (2 \, {\left (b^{5} d^{2} + 2 \, b^{3} d^{3} + b d^{4}\right )} x^{3} - {\left (4 \, b^{5} d + 11 \, b^{3} d^{2} + 7 \, b d^{3}\right )} x\right )} e^{\left (b^{2} x^{2} + d x^{2} + c\right )}}{8 \, \pi {\left (b^{6} d^{3} + 3 \, b^{4} d^{4} + 3 \, b^{2} d^{5} + d^{6}\right )}} \]
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\[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=e^{c} \int x^{5} e^{d x^{2}} \operatorname {erfi}{\left (b x \right )}\, dx \]
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\[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=\int { x^{5} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
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\[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=\int { x^{5} \operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
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Time = 5.30 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.90 \[ \int e^{c+d x^2} x^5 \text {erfi}(b x) \, dx=\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{d\,x^2+c}}{d^3}-\frac {x^2\,{\mathrm {e}}^{d\,x^2+c}}{d^2}+\frac {x^4\,{\mathrm {e}}^{d\,x^2+c}}{2\,d}\right )-\frac {b\,\mathrm {erfi}\left (x\,\sqrt {b^2+d}\right )\,{\mathrm {e}}^c}{2\,d^2\,{\left (b^2+d\right )}^{3/2}}-\frac {b\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {-b^2-d}\right )}{d^3\,\sqrt {-b^2-d}}+\frac {b\,x\,{\mathrm {e}}^{b^2\,x^2+d\,x^2+c}}{d^2\,\sqrt {\pi }\,\left (b^2+d\right )}+\frac {b\,x^5\,{\mathrm {e}}^c\,\left (\frac {3\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-x^2\,\left (b^2+d\right )}\right )}{4}+{\mathrm {e}}^{b^2\,x^2+d\,x^2}\,\left (\frac {3\,\sqrt {-x^2\,\left (b^2+d\right )}}{2}+{\left (-x^2\,\left (b^2+d\right )\right )}^{3/2}\right )-\frac {3\,\sqrt {\pi }}{4}\right )}{2\,d\,\sqrt {\pi }\,{\left (-x^2\,\left (b^2+d\right )\right )}^{5/2}} \]
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