Optimal. Leaf size=119 \[ \frac {3 e^c x^2}{4 b^3 \sqrt {\pi }}-\frac {e^c x^4}{4 b \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {Erf}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {Erf}(b x)}{2 b^2}+\frac {3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{4 b^3 \sqrt {\pi }} \]
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Rubi [A]
time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {6520, 6511, 12,
30} \begin {gather*} \frac {3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{4 \sqrt {\pi } b^3}+\frac {3 e^c x^2}{4 \sqrt {\pi } b^3}+\frac {x^3 e^{b^2 x^2+c} \text {Erf}(b x)}{2 b^2}-\frac {3 x e^{b^2 x^2+c} \text {Erf}(b x)}{4 b^4}-\frac {e^c x^4}{4 \sqrt {\pi } b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 6511
Rule 6520
Rubi steps
\begin {align*} \int e^{c+b^2 x^2} x^4 \text {erf}(b x) \, dx &=\frac {e^{c+b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}-\frac {3 \int e^{c+b^2 x^2} x^2 \text {erf}(b x) \, dx}{2 b^2}-\frac {\int e^c x^3 \, dx}{b \sqrt {\pi }}\\ &=-\frac {3 e^{c+b^2 x^2} x \text {erf}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}+\frac {3 \int e^{c+b^2 x^2} \text {erf}(b x) \, dx}{4 b^4}+\frac {3 \int e^c x \, dx}{2 b^3 \sqrt {\pi }}-\frac {e^c \int x^3 \, dx}{b \sqrt {\pi }}\\ &=-\frac {e^c x^4}{4 b \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {erf}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}+\frac {3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{4 b^3 \sqrt {\pi }}+\frac {\left (3 e^c\right ) \int x \, dx}{2 b^3 \sqrt {\pi }}\\ &=\frac {3 e^c x^2}{4 b^3 \sqrt {\pi }}-\frac {e^c x^4}{4 b \sqrt {\pi }}-\frac {3 e^{c+b^2 x^2} x \text {erf}(b x)}{4 b^4}+\frac {e^{c+b^2 x^2} x^3 \text {erf}(b x)}{2 b^2}+\frac {3 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{4 b^3 \sqrt {\pi }}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 100, normalized size = 0.84 \begin {gather*} \frac {e^c \left (6 b^2 x^2-2 b^4 x^4+2 b e^{b^2 x^2} \sqrt {\pi } x \left (-3+2 b^2 x^2\right ) \text {Erf}(b x)+3 \pi \text {Erf}(b x) \text {Erfi}(b x)-6 b^2 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )\right )}{8 b^5 \sqrt {\pi }} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{b^{2} x^{2}+c} x^{4} \erf \left (b x \right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erf}\left (b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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