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

3.1.90.1 Optimal result

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

output

-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/P 
i^(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)*erf(b*x+ 
a),x)/d^2
 

3.1.90.2 Mathematica [N/A]

Not integrable

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

input

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

output

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

3.1.90.3 Rubi [N/A]

Not integrable

Time = 3.75 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {6939, 2671, 2670, 2664, 2634, 2671, 2664, 2634, 2670, 2664, 2634, 6939, 2670, 2664, 2634, 6933}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6939

\(\displaystyle -\frac {b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^3dx}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erf}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2671

\(\displaystyle -\frac {b \left (\frac {\int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^2dx}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erf}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2670

\(\displaystyle -\frac {b \left (\frac {-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^2dx}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erf}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2664

\(\displaystyle -\frac {b \left (\frac {-\frac {a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^2dx}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erf}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^2dx}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erf}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2671

\(\displaystyle -\frac {b \left (-\frac {a b \left (\frac {\int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{2 \left (b^2-d\right )}-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erf}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2664

\(\displaystyle -\frac {b \left (-\frac {a b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}+\frac {e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{2 \left (b^2-d\right )}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erf}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {b \left (-\frac {a b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erf}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2670

\(\displaystyle -\frac {b \left (-\frac {a b \left (-\frac {a b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erf}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2664

\(\displaystyle -\frac {b \left (-\frac {a b \left (-\frac {a b \left (-\frac {a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erf}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {3 \int e^{d x^2+c} x^2 \text {erf}(a+b x)dx}{2 d}-\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 6939

\(\displaystyle -\frac {3 \left (-\frac {b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} \text {erf}(a+b x)dx}{2 d}+\frac {x e^{c+d x^2} \text {erf}(a+b x)}{2 d}\right )}{2 d}-\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2670

\(\displaystyle -\frac {3 \left (-\frac {b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} \text {erf}(a+b x)dx}{2 d}+\frac {x e^{c+d x^2} \text {erf}(a+b x)}{2 d}\right )}{2 d}-\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2664

\(\displaystyle -\frac {3 \left (-\frac {b \left (-\frac {a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} \text {erf}(a+b x)dx}{2 d}+\frac {x e^{c+d x^2} \text {erf}(a+b x)}{2 d}\right )}{2 d}-\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {3 \left (-\frac {\int e^{d x^2+c} \text {erf}(a+b x)dx}{2 d}-\frac {b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x e^{c+d x^2} \text {erf}(a+b x)}{2 d}\right )}{2 d}-\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

\(\Big \downarrow \) 6933

\(\displaystyle -\frac {3 \left (-\frac {\int e^{d x^2+c} \text {erf}(a+b x)dx}{2 d}-\frac {b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x e^{c+d x^2} \text {erf}(a+b x)}{2 d}\right )}{2 d}-\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erf}(a+b x)}{2 d}\)

input

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

output

$Aborted
 

3.1.90.3.1 Defintions of rubi rules used

rule 2634

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 2664

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[F^(a - b^2/ 
(4*c))   Int[F^((b + 2*c*x)^2/(4*c)), x], x] /; FreeQ[{F, a, b, c}, x]
 

rule 2670

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol 
] :> Simp[e*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] - Simp[(b*e - 2*c*d)/(2* 
c)   Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ 
[b*e - 2*c*d, 0]
 

rule 2671

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] + 
 (-Simp[(b*e - 2*c*d)/(2*c)   Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2), x] 
, x] - Simp[(m - 1)*(e^2/(2*c*Log[F]))   Int[(d + e*x)^(m - 2)*F^(a + b*x + 
 c*x^2), x], x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && 
GtQ[m, 1]
 

rule 6933

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Un 
integrable[E^(c + d*x^2)*Erf[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
 

rule 6939

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] : 
> Simp[x^(m - 1)*E^(c + d*x^2)*(Erf[a + b*x]/(2*d)), x] + (-Simp[(m - 1)/(2 
*d)   Int[x^(m - 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Simp[b/(d*Sqrt[Pi] 
)   Int[x^(m - 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; FreeQ[ 
{a, b, c, d}, x] && IGtQ[m, 1]
 

3.1.90.4 Maple [N/A] (verified)

Not integrable

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

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

input

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

output

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

3.1.90.5 Fricas [N/A]

Not integrable

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

input

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

output

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

3.1.90.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.1.90.7 Maxima [N/A]

Not integrable

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

input

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

output

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

3.1.90.8 Giac [N/A]

Not integrable

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

input

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

output

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

3.1.90.9 Mupad [N/A]

Not integrable

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

input

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

output

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