Optimal. Leaf size=527 \[ \frac {3 \text {Int}\left (e^{c+d x^2} \text {erfc}(a+b x),x\right )}{4 d^2}+\frac {3 a b^2 e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 d^2 \left (b^2-d\right )^{3/2}}+\frac {3 b e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{4 \sqrt {\pi } d^2 \left (b^2-d\right )}-\frac {3 a b^2 e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 d \left (b^2-d\right )^{5/2}}+\frac {a b^2 x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt {\pi } d \left (b^2-d\right )^2}-\frac {b x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt {\pi } d \left (b^2-d\right )}-\frac {b e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt {\pi } d \left (b^2-d\right )^2}-\frac {a^2 b^3 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt {\pi } d \left (b^2-d\right )^3}-\frac {a^3 b^4 e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d \left (b^2-d\right )^{7/2}}-\frac {3 x e^{c+d x^2} \text {erfc}(a+b x)}{4 d^2}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d} \]
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Rubi [A] time = 0.91, 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 = {} \[ \int e^{c+d x^2} x^4 \text {Erfc}(a+b x) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx &=\frac {e^{c+d x^2} x^3 \text {erfc}(a+b x)}{2 d}-\frac {3 \int e^{c+d x^2} x^2 \text {erfc}(a+b x) \, dx}{2 d}+\frac {b \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} x^3 \, dx}{d \sqrt {\pi }}\\ &=-\frac {b e^{-a^2+c-2 a b x-\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 {erfc}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(a+b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfc}(a+b x) \, dx}{4 d^2}-\frac {(3 b) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} x \, dx}{2 d^2 \sqrt {\pi }}+\frac {b \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} x \, dx}{\left (b^2-d\right ) d \sqrt {\pi }}-\frac {\left (a b^2\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} x^2 \, dx}{\left (b^2-d\right ) d \sqrt {\pi }}\\ &=\frac {3 b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\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 {erfc}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(a+b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfc}(a+b x) \, dx}{4 d^2}+\frac {\left (3 a b^2\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{2 \left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {\left (a b^2\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {\left (a b^2\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{\left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {\left (a^2 b^3\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} x \, dx}{\left (b^2-d\right )^2 d \sqrt {\pi }}\\ &=\frac {3 b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {a^2 b^3 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^3 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\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 {erfc}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(a+b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfc}(a+b x) \, dx}{4 d^2}-\frac {\left (a^3 b^4\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{\left (b^2-d\right )^3 d \sqrt {\pi }}+\frac {\left (3 a b^2 e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{2 \left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {\left (a b^2 e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {\left (a b^2 e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{\left (b^2-d\right )^2 d \sqrt {\pi }}\\ &=\frac {3 b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {a^2 b^3 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^3 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}+\frac {3 a b^2 e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2} d^2}-\frac {3 a b^2 e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{5/2} d}-\frac {3 e^{c+d x^2} x \text {erfc}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(a+b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfc}(a+b x) \, dx}{4 d^2}-\frac {\left (a^3 b^4 e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{\left (b^2-d\right )^3 d \sqrt {\pi }}\\ &=\frac {3 b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {a^2 b^3 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^3 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}+\frac {3 a b^2 e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2} d^2}-\frac {a^3 b^4 e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{7/2} d}-\frac {3 a b^2 e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{5/2} d}-\frac {3 e^{c+d x^2} x \text {erfc}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(a+b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfc}(a+b x) \, dx}{4 d^2}\\ \end {align*}
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Mathematica [A] time = 1.20, size = 0, normalized size = 0.00 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (x^{4} \operatorname {erf}\left (b x + a\right ) - x^{4}\right )} e^{\left (d x^{2} + c\right )}, x\right ) \]
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 \[ \int x^{4} \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d \,x^{2}+c} x^{4} \mathrm {erfc}\left (b x +a \right )\, dx \]
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 \[ \int x^{4} \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,\mathrm {erfc}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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