Optimal. Leaf size=97 \[ \frac {5 e^c \text {erfi}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}+\frac {x^2 e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^2}-\frac {e^{b^2 x^2+c} \text {erfi}(b x)}{2 b^4}-\frac {x e^{2 b^2 x^2+c}}{4 \sqrt {\pi } b^3} \]
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Rubi [A] time = 0.11, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {6387, 6384, 2204, 2212} \[ \frac {x^2 e^{b^2 x^2+c} \text {Erfi}(b x)}{2 b^2}-\frac {e^{b^2 x^2+c} \text {Erfi}(b x)}{2 b^4}+\frac {5 e^c \text {Erfi}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}-\frac {x e^{2 b^2 x^2+c}}{4 \sqrt {\pi } b^3} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2212
Rule 6384
Rule 6387
Rubi steps
\begin {align*} \int e^{c+b^2 x^2} x^3 \text {erfi}(b x) \, dx &=\frac {e^{c+b^2 x^2} x^2 \text {erfi}(b x)}{2 b^2}-\frac {\int e^{c+b^2 x^2} x \text {erfi}(b x) \, dx}{b^2}-\frac {\int e^{c+2 b^2 x^2} x^2 \, dx}{b \sqrt {\pi }}\\ &=-\frac {e^{c+2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erfi}(b x)}{2 b^2}+\frac {\int e^{c+2 b^2 x^2} \, dx}{4 b^3 \sqrt {\pi }}+\frac {\int e^{c+2 b^2 x^2} \, dx}{b^3 \sqrt {\pi }}\\ &=-\frac {e^{c+2 b^2 x^2} x}{4 b^3 \sqrt {\pi }}-\frac {e^{c+b^2 x^2} \text {erfi}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erfi}(b x)}{2 b^2}+\frac {5 e^c \text {erfi}\left (\sqrt {2} b x\right )}{8 \sqrt {2} b^4}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 0.79 \[ \frac {e^c \left (8 \sqrt {\pi } e^{b^2 x^2} \left (b^2 x^2-1\right ) \text {erfi}(b x)-4 b x e^{2 b^2 x^2}+5 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} b x\right )\right )}{16 \sqrt {\pi } b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 82, normalized size = 0.85 \[ -\frac {4 \, \sqrt {\pi } b^{2} x e^{\left (2 \, b^{2} x^{2} + c\right )} - 5 \, \sqrt {2} \pi \sqrt {b^{2}} \operatorname {erfi}\left (\sqrt {2} \sqrt {b^{2}} x\right ) e^{c} - 8 \, {\left (\pi b^{3} x^{2} - \pi b\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{16 \, \pi b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{b^{2} x^{2}+c} x^{3} \erfi \left (b x \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 117, normalized size = 1.21 \[ \frac {\sqrt {2}\,{\mathrm {e}}^c\,\mathrm {erfi}\left (\sqrt {2}\,x\,\sqrt {b^2}\right )}{16\,b\,{\left (b^2\right )}^{3/2}}-\frac {x\,{\mathrm {e}}^{2\,b^2\,x^2+c}}{4\,b^3\,\sqrt {\pi }}-\mathrm {erfi}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^4}-\frac {x^2\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}\right )-\frac {\sqrt {2}\,\mathrm {erf}\left (\sqrt {2}\,x\,\sqrt {-b^2}\right )\,{\mathrm {e}}^c}{4\,b\,{\left (-b^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int x^{3} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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