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

Optimal. Leaf size=185 \[ \frac{3 \text{Unintegrable}\left (\text{Erfc}(b x) e^{c+d x^2},x\right )}{4 d^2}+\frac{3 b e^{c-x^2 \left (b^2-d\right )}}{4 \sqrt{\pi } d^2 \left (b^2-d\right )}-\frac{b x^2 e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt{\pi } d \left (b^2-d\right )}-\frac{b e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt{\pi } d \left (b^2-d\right )^2}-\frac{3 x \text{Erfc}(b x) e^{c+d x^2}}{4 d^2}+\frac{x^3 \text{Erfc}(b x) e^{c+d x^2}}{2 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.233365, 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 = {} \[ \int e^{c+d x^2} x^4 \text{Erfc}(b x) \, dx \]

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*}

Mathematica [A]  time = 0.757467, size = 0, normalized size = 0. \[ \int e^{c+d x^2} x^4 \text{Erfc}(b x) \, dx \]

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]

________________________________________________________________________________________

Maple [A]  time = 0.141, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}{x}^{4}{\it erfc} \left ( bx \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),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{erfc}\left (b x\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),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (x^{4} \operatorname{erf}\left (b x\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),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{erfc}\left (b x\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),x, algorithm="giac")

[Out]

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