3.2.63 \(\int e^{c+d x^2} x^2 \text {Erfc}(b x) \, dx\) [163]

Optimal. Leaf size=84 \[ -\frac {b e^{c-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right ) d \sqrt {\pi }}+\frac {e^{c+d x^2} x \text {Erfc}(b x)}{2 d}-\frac {\text {Int}\left (e^{c+d x^2} \text {Erfc}(b x),x\right )}{2 d} \]

[Out]

1/2*exp(d*x^2+c)*x*erfc(b*x)/d-1/2*b*exp(c-(b^2-d)*x^2)/(b^2-d)/d/Pi^(1/2)-1/2*Unintegrable(exp(d*x^2+c)*erfc(
b*x),x)/d

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Rubi [A]
time = 0.06, 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 = {} \begin {gather*} \int e^{c+d x^2} x^2 \text {Erfc}(b x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(c + d*x^2)*x^2*Erfc[b*x],x]

[Out]

-1/2*(b*E^(c - (b^2 - d)*x^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*}

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Mathematica [A]
time = 0.45, size = 0, normalized size = 0.00 \begin {gather*} \int e^{c+d x^2} x^2 \text {Erfc}(b x) \, dx \end {gather*}

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 = 0.14, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{d \,x^{2}+c} x^{2} \mathrm {erfc}\left (b x \right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d*x^2+c)*x^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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{c} \int x^{2} e^{d x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*exp(c + d*x^2)*erfc(b*x),x)

[Out]

int(x^2*exp(c + d*x^2)*erfc(b*x), x)

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