Optimal. Leaf size=79 \[ \frac {e^c x}{\sqrt {\pi } b^3}+\frac {x^2 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{2 b^4}-\frac {e^c x^3}{3 \sqrt {\pi } b} \]
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Rubi [A] time = 0.08, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6385, 6382, 8, 12, 30} \[ \frac {x^2 e^{b^2 x^2+c} \text {Erf}(b x)}{2 b^2}-\frac {e^{b^2 x^2+c} \text {Erf}(b x)}{2 b^4}+\frac {e^c x}{\sqrt {\pi } b^3}-\frac {e^c x^3}{3 \sqrt {\pi } b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 30
Rule 6382
Rule 6385
Rubi steps
\begin {align*} \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx &=\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}-\frac {\int e^{c+b^2 x^2} x \text {erf}(b x) \, dx}{b^2}-\frac {\int e^c x^2 \, dx}{b \sqrt {\pi }}\\ &=-\frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}+\frac {\int e^c \, dx}{b^3 \sqrt {\pi }}-\frac {e^c \int x^2 \, dx}{b \sqrt {\pi }}\\ &=\frac {e^c x}{b^3 \sqrt {\pi }}-\frac {e^c x^3}{3 b \sqrt {\pi }}-\frac {e^{c+b^2 x^2} \text {erf}(b x)}{2 b^4}+\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 0.72 \[ \frac {e^c \left (-2 b^3 x^3+3 \sqrt {\pi } e^{b^2 x^2} \left (b^2 x^2-1\right ) \text {erf}(b x)+6 b x\right )}{6 \sqrt {\pi } b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 55, normalized size = 0.70 \[ -\frac {3 \, {\left (\pi - \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} + 2 \, \sqrt {\pi } {\left (b^{3} x^{3} - 3 \, b x\right )} e^{c}}{6 \, \pi b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 71, normalized size = 0.90 \[ \frac {1}{2} \, {\left (\frac {{\left (b^{2} x^{2} + c - 1\right )} e^{\left (b^{2} x^{2} + c\right )}}{b^{4}} - \frac {c e^{\left (b^{2} x^{2} + c\right )}}{b^{4}}\right )} \operatorname {erf}\left (b x\right ) - \frac {b^{2} x^{3} e^{c} - 3 \, x e^{c}}{3 \, \sqrt {\pi } b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 66, normalized size = 0.84 \[ \frac {\frac {\erf \left (b x \right ) {\mathrm e}^{c} \left (\frac {b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2}\right )}{b^{3}}-\frac {{\mathrm e}^{c} \left (\frac {1}{3} b^{3} x^{3}-b x \right )}{\sqrt {\pi }\, b^{3}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 59, normalized size = 0.75 \[ -\frac {2 \, b^{3} x^{3} e^{c} - 3 \, {\left (\sqrt {\pi } b^{2} x^{2} e^{c} - \sqrt {\pi } e^{c}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - 6 \, b x e^{c}}{6 \, \sqrt {\pi } b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 65, normalized size = 0.82 \[ \frac {3\,x\,{\mathrm {e}}^c-b^2\,x^3\,{\mathrm {e}}^c}{3\,b^3\,\sqrt {\pi }}-\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^4}-\frac {x^2\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 131.39, size = 76, normalized size = 0.96 \[ \begin {cases} - \frac {x^{3} e^{c}}{3 \sqrt {\pi } b} + \frac {x^{2} e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{2}} + \frac {x e^{c}}{\sqrt {\pi } b^{3}} - \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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