3.162 \(\int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx\)

Optimal. Leaf size=186 \[ \frac {3 \text {Int}\left (\text {erfc}(b x) e^{c+d x^2},x\right )}{4 d^2}+\frac {3 b e^{c-x^2 \left (b^2-d\right )}}{4 \sqrt {\pi } d^2 \left (b^2-d\right )}-\frac {b x^2 e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt {\pi } d \left (b^2-d\right )}-\frac {b e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt {\pi } d \left (b^2-d\right )^2}-\frac {3 x \text {erfc}(b x) e^{c+d x^2}}{4 d^2}+\frac {x^3 \text {erfc}(b x) e^{c+d x^2}}{2 d} \]

[Out]

-3/4*exp(d*x^2+c)*x*erfc(b*x)/d^2+1/2*exp(d*x^2+c)*x^3*erfc(b*x)/d+3/4*b*exp(c-(b^2-d)*x^2)/(b^2-d)/d^2/Pi^(1/
2)-1/2*b*exp(c-(b^2-d)*x^2)/(b^2-d)^2/d/Pi^(1/2)-1/2*b*exp(c-(b^2-d)*x^2)*x^2/(b^2-d)/d/Pi^(1/2)+3/4*Unintegra
ble(exp(d*x^2+c)*erfc(b*x),x)/d^2

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Rubi [A]  time = 0.23, 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 = {} \[ \int e^{c+d x^2} x^4 \text {Erfc}(b x) \, dx \]

Verification is Not applicable to the result.

[In]

Int[E^(c + d*x^2)*x^4*Erfc[b*x],x]

[Out]

(3*b*E^(c - (b^2 - d)*x^2))/(4*(b^2 - d)*d^2*Sqrt[Pi]) - (b*E^(c - (b^2 - d)*x^2))/(2*(b^2 - d)^2*d*Sqrt[Pi])
- (b*E^(c - (b^2 - d)*x^2)*x^2)/(2*(b^2 - d)*d*Sqrt[Pi]) - (3*E^(c + d*x^2)*x*Erfc[b*x])/(4*d^2) + (E^(c + d*x
^2)*x^3*Erfc[b*x])/(2*d) + (3*Defer[Int][E^(c + d*x^2)*Erfc[b*x], x])/(4*d^2)

Rubi steps

\begin {align*} \int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx &=\frac {e^{c+d x^2} x^3 \text {erfc}(b x)}{2 d}-\frac {3 \int e^{c+d x^2} x^2 \text {erfc}(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 {erfc}(b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfc}(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 {erfc}(b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfc}(b x) \, dx}{4 d^2}\\ \end {align*}

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Mathematica [A]  time = 0.95, size = 0, normalized size = 0.00 \[ \int e^{c+d x^2} x^4 \text {erfc}(b x) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[E^(c + d*x^2)*x^4*Erfc[b*x],x]

[Out]

Integrate[E^(c + d*x^2)*x^4*Erfc[b*x], x]

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (x^{4} \operatorname {erf}\left (b x\right ) - x^{4}\right )} e^{\left (d x^{2} + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*x^4*erfc(b*x),x, algorithm="fricas")

[Out]

integral(-(x^4*erf(b*x) - x^4)*e^(d*x^2 + c), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*x^4*erfc(b*x),x, algorithm="giac")

[Out]

integrate(x^4*erfc(b*x)*e^(d*x^2 + c), x)

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maple [A]  time = 0.10, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d \,x^{2}+c} x^{4} \mathrm {erfc}\left (b x \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d*x^2+c)*x^4*erfc(b*x),x)

[Out]

int(exp(d*x^2+c)*x^4*erfc(b*x),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*x^4*erfc(b*x),x, algorithm="maxima")

[Out]

integrate(x^4*erfc(b*x)*e^(d*x^2 + c), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*exp(c + d*x^2)*erfc(b*x),x)

[Out]

int(x^4*exp(c + d*x^2)*erfc(b*x), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int x^{4} e^{d x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x**2+c)*x**4*erfc(b*x),x)

[Out]

exp(c)*Integral(x**4*exp(d*x**2)*erfc(b*x), x)

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