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

Optimal. Leaf size=83 \[ -\frac{\text{Unintegrable}\left (\text{Erfc}(b x) e^{c+d x^2},x\right )}{2 d}-\frac{b e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt{\pi } d \left (b^2-d\right )}+\frac{x \text{Erfc}(b x) e^{c+d x^2}}{2 d} \]

[Out]

-(b*E^(c - (b^2 - d)*x^2))/(2*(b^2 - d)*d*Sqrt[Pi]) + (E^(c + d*x^2)*x*Erfc[b*x])/(2*d) - Unintegrable[E^(c +
d*x^2)*Erfc[b*x], x]/(2*d)

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Rubi [A]  time = 0.0936751, 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 = {} \[ \int e^{c+d x^2} x^2 \text{Erfc}(b x) \, dx \]

Verification is Not applicable to the result.

[In]

Int[E^(c + d*x^2)*x^2*Erfc[b*x],x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.589352, size = 0, normalized size = 0. \[ \int e^{c+d x^2} x^2 \text{Erfc}(b x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.241, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}{x}^{2}{\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^2*erfc(b*x),x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \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^2*erfc(b*x),x, algorithm="maxima")

[Out]

integrate(x^2*erfc(b*x)*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^{2} \operatorname{erf}\left (b x\right ) - x^{2}\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^2*erfc(b*x),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \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^2*erfc(b*x),x, algorithm="giac")

[Out]

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