\(\int e^{c+d x^2} x^4 \text {erf}(a+b x) \, dx\) [90]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int e^{c+d x^2} x^4 \text {erf}(a+b x) \, dx=-\frac {3 b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}+\frac {a^2 b^3 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^3 d \sqrt {\pi }}+\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erf}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erf}(a+b x)}{2 d}-\frac {3 a b^2 e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2} d^2}+\frac {a^3 b^4 e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{7/2} d}+\frac {3 a b^2 e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{5/2} d}+\frac {3 \text {Int}\left (e^{c+d x^2} \text {erf}(a+b x),x\right )}{4 d^2} \]

[Out]

-3/4*exp(d*x^2+c)*x*erf(b*x+a)/d^2+1/2*exp(d*x^2+c)*x^3*erf(b*x+a)/d-3/4*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^2+1/2*a^3*b^4*exp(c+a^2*d/(b^2-d))*erf((a*b+(b^2-d)*x)/(b^2-d)^(1/2))
/(b^2-d)^(7/2)/d+3/4*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)^(5/2)/d-3/4*b*exp(-
a^2+c-2*a*b*x-(b^2-d)*x^2)/(b^2-d)/d^2/Pi^(1/2)+1/2*a^2*b^3*exp(-a^2+c-2*a*b*x-(b^2-d)*x^2)/(b^2-d)^3/d/Pi^(1/
2)+1/2*b*exp(-a^2+c-2*a*b*x-(b^2-d)*x^2)/(b^2-d)^2/d/Pi^(1/2)-1/2*a*b^2*exp(-a^2+c-2*a*b*x-(b^2-d)*x^2)*x/(b^2
-d)^2/d/Pi^(1/2)+1/2*b*exp(-a^2+c-2*a*b*x-(b^2-d)*x^2)*x^2/(b^2-d)/d/Pi^(1/2)+3/4*Unintegrable(exp(d*x^2+c)*er
f(b*x+a),x)/d^2

Rubi [N/A]

Not integrable

Time = 0.72 (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 = {} \[ \int e^{c+d x^2} x^4 \text {erf}(a+b x) \, dx=\int e^{c+d x^2} x^4 \text {erf}(a+b x) \, dx \]

[In]

Int[E^(c + d*x^2)*x^4*Erf[a + b*x],x]

[Out]

(-3*b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(4*(b^2 - d)*d^2*Sqrt[Pi]) + (a^2*b^3*E^(-a^2 + c - 2*a*b*x - (b
^2 - d)*x^2))/(2*(b^2 - d)^3*d*Sqrt[Pi]) + (b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(2*(b^2 - d)^2*d*Sqrt[Pi
]) - (a*b^2*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2)*x)/(2*(b^2 - d)^2*d*Sqrt[Pi]) + (b*E^(-a^2 + c - 2*a*b*x -
(b^2 - d)*x^2)*x^2)/(2*(b^2 - d)*d*Sqrt[Pi]) - (3*E^(c + d*x^2)*x*Erf[a + b*x])/(4*d^2) + (E^(c + d*x^2)*x^3*E
rf[a + b*x])/(2*d) - (3*a*b^2*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(4*(b^2 - d)^(
3/2)*d^2) + (a^3*b^4*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(2*(b^2 - d)^(7/2)*d) +
(3*a*b^2*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/(4*(b^2 - d)^(5/2)*d) + (3*Defer[I
nt][E^(c + d*x^2)*Erf[a + b*x], x])/(4*d^2)

Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+d x^2} x^3 \text {erf}(a+b x)}{2 d}-\frac {3 \int e^{c+d x^2} 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^3 \, dx}{d \sqrt {\pi }} \\ & = \frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erf}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erf}(a+b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erf}(a+b x) \, dx}{4 d^2}+\frac {(3 b) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} x \, dx}{2 d^2 \sqrt {\pi }}-\frac {b \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} x \, dx}{\left (b^2-d\right ) d \sqrt {\pi }}+\frac {\left (a b^2\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} x^2 \, dx}{\left (b^2-d\right ) d \sqrt {\pi }} \\ & = -\frac {3 b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \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 )^2 d \sqrt {\pi }}-\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erf}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erf}(a+b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erf}(a+b x) \, dx}{4 d^2}-\frac {\left (3 a b^2\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{2 \left (b^2-d\right ) d^2 \sqrt {\pi }}+\frac {\left (a b^2\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\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 )^2 d \sqrt {\pi }}-\frac {\left (a^2 b^3\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} x \, dx}{\left (b^2-d\right )^2 d \sqrt {\pi }} \\ & = -\frac {3 b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}+\frac {a^2 b^3 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^3 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 )^2 d \sqrt {\pi }}-\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erf}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erf}(a+b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erf}(a+b x) \, dx}{4 d^2}+\frac {\left (a^3 b^4\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{\left (b^2-d\right )^3 d \sqrt {\pi }}-\frac {\left (3 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}{2 \left (b^2-d\right ) d^2 \sqrt {\pi }}+\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}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\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 )^2 d \sqrt {\pi }} \\ & = -\frac {3 b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}+\frac {a^2 b^3 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^3 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 )^2 d \sqrt {\pi }}-\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erf}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erf}(a+b x)}{2 d}-\frac {3 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 )}{4 \left (b^2-d\right )^{3/2} d^2}+\frac {3 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 )}{4 \left (b^2-d\right )^{5/2} d}+\frac {3 \int e^{c+d x^2} \text {erf}(a+b x) \, dx}{4 d^2}+\frac {\left (a^3 b^4 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 )^3 d \sqrt {\pi }} \\ & = -\frac {3 b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}+\frac {a^2 b^3 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^3 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 )^2 d \sqrt {\pi }}-\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erf}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erf}(a+b x)}{2 d}-\frac {3 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 )}{4 \left (b^2-d\right )^{3/2} d^2}+\frac {a^3 b^4 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 )^{7/2} d}+\frac {3 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 )}{4 \left (b^2-d\right )^{5/2} d}+\frac {3 \int e^{c+d x^2} \text {erf}(a+b x) \, dx}{4 d^2} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int e^{c+d x^2} x^4 \text {erf}(a+b x) \, dx=\int e^{c+d x^2} x^4 \text {erf}(a+b x) \, dx \]

[In]

Integrate[E^(c + d*x^2)*x^4*Erf[a + b*x],x]

[Out]

Integrate[E^(c + d*x^2)*x^4*Erf[a + b*x], x]

Maple [N/A] (verified)

Not integrable

Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

\[\int {\mathrm e}^{d \,x^{2}+c} x^{4} \operatorname {erf}\left (b x +a \right )d x\]

[In]

int(exp(d*x^2+c)*x^4*erf(b*x+a),x)

[Out]

int(exp(d*x^2+c)*x^4*erf(b*x+a),x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erf}(a+b x) \, dx=\int { x^{4} \operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[In]

integrate(exp(d*x^2+c)*x^4*erf(b*x+a),x, algorithm="fricas")

[Out]

integral(x^4*erf(b*x + a)*e^(d*x^2 + c), x)

Sympy [F(-1)]

Timed out. \[ \int e^{c+d x^2} x^4 \text {erf}(a+b x) \, dx=\text {Timed out} \]

[In]

integrate(exp(d*x**2+c)*x**4*erf(b*x+a),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erf}(a+b x) \, dx=\int { x^{4} \operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[In]

integrate(exp(d*x^2+c)*x^4*erf(b*x+a),x, algorithm="maxima")

[Out]

integrate(x^4*erf(b*x + a)*e^(d*x^2 + c), x)

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erf}(a+b x) \, dx=\int { x^{4} \operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[In]

integrate(exp(d*x^2+c)*x^4*erf(b*x+a),x, algorithm="giac")

[Out]

integrate(x^4*erf(b*x + a)*e^(d*x^2 + c), x)

Mupad [N/A]

Not integrable

Time = 6.86 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erf}(a+b x) \, dx=\int x^4\,\mathrm {erf}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c} \,d x \]

[In]

int(x^4*erf(a + b*x)*exp(c + d*x^2),x)

[Out]

int(x^4*erf(a + b*x)*exp(c + d*x^2), x)