3.193 \(\int e^{c+d x^2} x^4 \text{Erfc}(a+b x) \, dx\)

Optimal. Leaf size=526 \[ \frac{3 \text{Unintegrable}\left (e^{c+d x^2} \text{Erfc}(a+b x),x\right )}{4 d^2}+\frac{3 a b^2 e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{4 d^2 \left (b^2-d\right )^{3/2}}+\frac{3 b e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{4 \sqrt{\pi } d^2 \left (b^2-d\right )}-\frac{a^3 b^4 e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{2 d \left (b^2-d\right )^{7/2}}-\frac{3 a b^2 e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{4 d \left (b^2-d\right )^{5/2}}-\frac{a^2 b^3 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt{\pi } d \left (b^2-d\right )^3}+\frac{a b^2 x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt{\pi } d \left (b^2-d\right )^2}-\frac{b x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt{\pi } d \left (b^2-d\right )}-\frac{b e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt{\pi } d \left (b^2-d\right )^2}-\frac{3 x e^{c+d x^2} \text{Erfc}(a+b x)}{4 d^2}+\frac{x^3 e^{c+d x^2} \text{Erfc}(a+b x)}{2 d} \]

[Out]

(3*b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(4*(b^2 - d)*d^2*Sqrt[Pi]) - (a^2*b^3*E^(-a^2 + c - 2*a*b*x - (b^
2 - d)*x^2))/(2*(b^2 - d)^3*d*Sqrt[Pi]) - (b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(2*(b^2 - d)^2*d*Sqrt[Pi]
) + (a*b^2*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2)*x)/(2*(b^2 - d)^2*d*Sqrt[Pi]) - (b*E^(-a^2 + c - 2*a*b*x - (
b^2 - d)*x^2)*x^2)/(2*(b^2 - d)*d*Sqrt[Pi]) + (3*a*b^2*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[
b^2 - d]])/(4*(b^2 - d)^(3/2)*d^2) - (a^3*b^4*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]]
)/(2*(b^2 - d)^(7/2)*d) - (3*a*b^2*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(4*(b^2 -
d)^(5/2)*d) - (3*E^(c + d*x^2)*x*Erfc[a + b*x])/(4*d^2) + (E^(c + d*x^2)*x^3*Erfc[a + b*x])/(2*d) + (3*Uninte
grable[E^(c + d*x^2)*Erfc[a + b*x], x])/(4*d^2)

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Rubi [A]  time = 0.907243, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int e^{c+d x^2} x^4 \text{Erfc}(a+b x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(3*b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(4*(b^2 - d)*d^2*Sqrt[Pi]) - (a^2*b^3*E^(-a^2 + c - 2*a*b*x - (b^
2 - d)*x^2))/(2*(b^2 - d)^3*d*Sqrt[Pi]) - (b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(2*(b^2 - d)^2*d*Sqrt[Pi]
) + (a*b^2*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2)*x)/(2*(b^2 - d)^2*d*Sqrt[Pi]) - (b*E^(-a^2 + c - 2*a*b*x - (
b^2 - d)*x^2)*x^2)/(2*(b^2 - d)*d*Sqrt[Pi]) + (3*a*b^2*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[
b^2 - d]])/(4*(b^2 - d)^(3/2)*d^2) - (a^3*b^4*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]]
)/(2*(b^2 - d)^(7/2)*d) - (3*a*b^2*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(4*(b^2 -
d)^(5/2)*d) - (3*E^(c + d*x^2)*x*Erfc[a + b*x])/(4*d^2) + (E^(c + d*x^2)*x^3*Erfc[a + b*x])/(2*d) + (3*Defer[
Int][E^(c + d*x^2)*Erfc[a + b*x], x])/(4*d^2)

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*}

Mathematica [A]  time = 1.04229, size = 0, normalized size = 0. \[ \int e^{c+d x^2} x^4 \text{Erfc}(a+b x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.146, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}{x}^{4}{\it erfc} \left ( bx+a \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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