Optimal. Leaf size=186 \[ \frac {3 \text {Int}\left (\text {erf}(b x) e^{c+d x^2},x\right )}{4 d^2}-\frac {3 b e^{c-x^2 \left (b^2-d\right )}}{4 \sqrt {\pi } d^2 \left (b^2-d\right )}+\frac {b x^2 e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt {\pi } d \left (b^2-d\right )}+\frac {b e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt {\pi } d \left (b^2-d\right )^2}-\frac {3 x \text {erf}(b x) e^{c+d x^2}}{4 d^2}+\frac {x^3 \text {erf}(b x) e^{c+d x^2}}{2 d} \]
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Rubi [A] time = 0.24, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int e^{c+d x^2} x^4 \text {Erf}(b x) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int e^{c+d x^2} x^4 \text {erf}(b x) \, dx &=\frac {e^{c+d x^2} x^3 \text {erf}(b x)}{2 d}-\frac {3 \int e^{c+d x^2} x^2 \text {erf}(b x) \, dx}{2 d}-\frac {b \int e^{c-\left (b^2-d\right ) x^2} x^3 \, dx}{d \sqrt {\pi }}\\ &=\frac {b e^{c-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erf}(b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erf}(b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erf}(b x) \, dx}{4 d^2}+\frac {(3 b) \int e^{c-\left (b^2-d\right ) x^2} x \, dx}{2 d^2 \sqrt {\pi }}-\frac {b \int e^{c+\left (-b^2+d\right ) x^2} x \, dx}{\left (b^2-d\right ) d \sqrt {\pi }}\\ &=-\frac {3 b e^{c-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}+\frac {b e^{c-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {b e^{c-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erf}(b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erf}(b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erf}(b x) \, dx}{4 d^2}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 0, normalized size = 0.00 \[ \int e^{c+d x^2} x^4 \text {erf}(b x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{4} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d \,x^{2}+c} x^{4} \erf \left (b x \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erf}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int x^{4} e^{d x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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