\(\int e^{c+d x^2} x^2 \text {erf}(a+b x) \, dx\) [91]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int e^{c+d x^2} x^2 \text {erf}(a+b x) \, dx=\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right ) d \sqrt {\pi }}+\frac {e^{c+d x^2} x \text {erf}(a+b x)}{2 d}+\frac {a b^2 e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2} d}-\frac {\text {Int}\left (e^{c+d x^2} \text {erf}(a+b x),x\right )}{2 d} \]

[Out]

1/2*exp(d*x^2+c)*x*erf(b*x+a)/d+1/2*a*b^2*exp(c+a^2*d/(b^2-d))*erf((a*b+(b^2-d)*x)/(b^2-d)^(1/2))/(b^2-d)^(3/2
)/d+1/2*b*exp(-a^2+c-2*a*b*x-(b^2-d)*x^2)/(b^2-d)/d/Pi^(1/2)-1/2*Unintegrable(exp(d*x^2+c)*erf(b*x+a),x)/d

Rubi [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int e^{c+d x^2} x^2 \text {erf}(a+b x) \, dx=\int e^{c+d x^2} x^2 \text {erf}(a+b x) \, dx \]

[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*} \text {integral}& = \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 [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int e^{c+d x^2} x^2 \text {erf}(a+b x) \, dx=\int e^{c+d x^2} x^2 \text {erf}(a+b x) \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 0.15 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

\[\int {\mathrm e}^{d \,x^{2}+c} x^{2} \operatorname {erf}\left (b x +a \right )d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^2 \text {erf}(a+b x) \, dx=\int { x^{2} \operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 55.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^2 \text {erf}(a+b x) \, dx=e^{c} \int x^{2} e^{d x^{2}} \operatorname {erf}{\left (a + b x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^2 \text {erf}(a+b x) \, dx=\int { x^{2} \operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^2 \text {erf}(a+b x) \, dx=\int { x^{2} \operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 6.83 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^2 \text {erf}(a+b x) \, dx=\int x^2\,\mathrm {erf}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c} \,d x \]

[In]

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

[Out]

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