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

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

[Out]

1/2*exp(d*x^2+c)*x*erf(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)*erf(b*
x),x)/d

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Rubi [A]  time = 0.09, 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 = {} \[ \int e^{c+d x^2} x^2 \text {Erf}(b x) \, dx \]

Verification is Not applicable to the result.

[In]

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

Rubi steps

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

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Mathematica [A]  time = 0.23, size = 0, normalized size = 0.00 \[ \int e^{c+d x^2} x^2 \text {erf}(b x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.22, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d \,x^{2}+c} x^{2} \erf \left (b x \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*exp(c + d*x^2)*erf(b*x),x)

[Out]

int(x^2*exp(c + d*x^2)*erf(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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