3.91 \(\int e^{c+d x^2} x^2 \text{Erf}(a+b x) \, dx\)

Optimal. Leaf size=163 \[ -\frac{\text{Unintegrable}\left (e^{c+d x^2} \text{Erf}(a+b x),x\right )}{2 d}+\frac{a b^2 e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )}{2 d \left (b^2-d\right )^{3/2}}+\frac{b e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt{\pi } d \left (b^2-d\right )}+\frac{x e^{c+d x^2} \text{Erf}(a+b x)}{2 d} \]

[Out]

(b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(2*(b^2 - d)*d*Sqrt[Pi]) + (E^(c + d*x^2)*x*Erf[a + b*x])/(2*d) + (
a*b^2*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*(b^2 - d)^(3/2)*d) - Unintegrable[E
^(c + d*x^2)*Erf[a + b*x], x]/(2*d)

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Rubi [A]  time = 0.189922, 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 = {} \[ \int e^{c+d x^2} x^2 \text{Erf}(a+b x) \, dx \]

Verification is Not applicable to the result.

[In]

Int[E^(c + d*x^2)*x^2*Erf[a + b*x],x]

[Out]

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

Rubi steps

\begin{align*} \int e^{c+d x^2} x^2 \text{erf}(a+b x) \, dx &=\frac{e^{c+d x^2} x \text{erf}(a+b x)}{2 d}-\frac{\int e^{c+d x^2} \text{erf}(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 \, dx}{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 ) d \sqrt{\pi }}+\frac{e^{c+d x^2} x \text{erf}(a+b x)}{2 d}-\frac{\int e^{c+d x^2} \text{erf}(a+b x) \, dx}{2 d}+\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 ) 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 ) d \sqrt{\pi }}+\frac{e^{c+d x^2} x \text{erf}(a+b x)}{2 d}-\frac{\int e^{c+d x^2} \text{erf}(a+b x) \, dx}{2 d}+\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 ) 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 ) d \sqrt{\pi }}+\frac{e^{c+d x^2} x \text{erf}(a+b x)}{2 d}+\frac{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 )}{2 \left (b^2-d\right )^{3/2} d}-\frac{\int e^{c+d x^2} \text{erf}(a+b x) \, dx}{2 d}\\ \end{align*}

Mathematica [A]  time = 0.39214, size = 0, normalized size = 0. \[ \int e^{c+d x^2} x^2 \text{Erf}(a+b x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.25, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}{x}^{2}{\it Erf} \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^2*erf(b*x+a),x)

[Out]

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

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

[Out]

integrate(x^2*erf(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 (x^{2} \operatorname{erf}\left (b x + a\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*erf(b*x+a),x, algorithm="fricas")

[Out]

integral(x^2*erf(b*x + a)*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**2*erf(b*x+a),x)

[Out]

Timed out

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

[Out]

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